Non-hydrostatic modeling of drag, inertia and porous effects in wave propagation over dense vegetation fields
Introduction
Coastal wetlands, such as mangroves, saltmarshes and seagrasses, are now widely recognized as effective buffers to incident wave energy, even during storm conditions (Asano et al., 1988; Arkema et al., 2013; Möller et al., 2014). Initiatives have now been taken to integrate these natural or constructed wetlands into overall coastal-protection schemes to mitigate wave impacts and associated erosion (Borsje et al., 2011; Temmerman et al., 2013). One example is the “Building with Nature” approach originated in the Netherlands, which includes natural wetlands in the infrastructure designs for improved flexibility and ecosystem services (Borsje et al., 2011; Cuc et al., 2015; de Vriend et al., 2015).
To design the required wetland space for wave dampening, quantitative assessment of the efficiency of wave damping by vegetation is needed (Bouma et al., 2014). Furthermore, besides wave energy dissipation, other wave propagation processes in vegetation wetlands, such as wave reflection and diffraction, should be properly quantified, since wave fields as a whole may have important impacts on an ecosystem's initial establishment and long-term health (Mariotti and Fagherazzi, 2010; Balke et al., 2011; Hu et al., 2015).
With continuous development, numerical models are becoming valuable tools to estimate wave propagation over coastal wetlands under various scenarios (Borsje et al., 2011; Mei et al., 2011; Suzuki et al., 2012; Liu et al., 2015; van Loon-Steensma et al., 2016). Several models that quantify the effect and process of wave propagation through vegetation fields are listed in Table 1. These models are categorized into two groups based on their controlling physical equations: energy-conservation models and momentum-conservation models. In the first group, wave dissipation by vegetation was initially modeled as added bottom friction (Hasselmann and Collins, 1968; Möller et al., 1999). Dalrymple et al. (1984) introduced a theoretical model that accounts for wave dissipation by vertical vegetation cylinders over the water column following linear wave theory. In this model, the wave dissipation in vegetation canopies is attributed to the drag force exerted by vegetation stems expressed by Morison equations (Morison et al., 1950). Several parameters describing vegetation canopies, such as number of vegetation stems in a unit area, stem diameter, and vegetation height, can be accounted for explicitly. Thus, the process of wave dissipation by vegetation can be quantified in detail. More recent work has modified the original model of Dalrymple et al. (1984) to incorporate the effect of wave breaking, wave irregularity and wave-current interaction (Méndez and Losada, 2004; Losada et al., 2016). Among energy-conservation models, the effect of wave dissipation by vegetation has also been introduced to a spectral-action balance equation model (Suzuki et al., 2012) and mild-slope equation models (Tang et al., 2015; Cao et al., 2015).
Momentum-conservation models seek to simulate wave propagation through vegetation by quantifying vegetation-induced momentum lost. The applied equations in these models include shallow water equations, Boussinesq-type equations, and Reynolds-averaged Navier-Stokes (RANS) equations (Table 1). As these equations are all based on momentum conservation, these models can provide not only wave-field information but also velocity structures with intra-wave (phase-resolving) resolutions, which is important for interpreting both wave propagation and sediment transport processes in coastal wetlands. Because of the increased scope and resolutions, these models are generally expensive in terms of computation time comparing to wave-energy conservation models.
Reviewing the previous work listed in Table 1, it becomes clear that drag force induced by horizontal vegetation stems/roots, inertia force, and porosity are often neglected in numerical models. Existing models often only consider vegetation structures as vertical cylinders, but natural vegetation systems such as mangroves have complex root systems composed of both vertical and horizontal roots (Ohira et al., 2013; Kamal et al., 2014) (Fig. 1a and b). Additionally, horizontal vegetation stems are seen in recent innovative coastal-protection projects that apply porous brushwood groins for wave damping and mangrove nurseries, e.g., in “Build with Nature, Indonesia” (Lucas, 2017). The force acting on horizontal vegetation cylinders has both horizontal and vertical components, whereas the latter is generally neglected in existing models that deal with only vertical vegetation cylinders (Fig. 2). The force on the horizontal vegetation stems/roots and the associated wave-energy dissipation need to be further investigated both theoretically and numerically.
Besides drag force, inertia force is the other component in the total force that acts on vegetation (Morison et al., 1950; Chen et al., 2018; Yao et al., 2018). Inertia force is commonly ignored in wave-vegetation modeling because the work done by the theoretical inertia force over one wave cycle is zero if linear wave theory is applied. However, coastal wetlands are normally located in shallow intertidal areas, where wave nonlinearity exists. For nonlinear waves, the work done by inertia force is nonzero. The effect of inertia on wave dissipation can be of greater importance in the case of dense vegetation with lower porosities, as inertia force is proportional to the spatial occupation of vegetation per unit volume (i.e., where and are the number of plants per square meter and the stem diameter, respectively). The value of can be as high as 0.45 in natural mangroves and even 0.65 in constructed coastal wetlands (Furukawa et al., 1997; Mazda et al., 1997; Serra et al., 2004). Thus, the inertia effect in these high-density conditions is potentially important in wave-propagation modeling, and is worthy of detailed investigation.
Also commonly ignored is the porosity effect, which is induced by the existence of vegetation in the water column that can “squeeze” the flow passing through it, leading to higher in-canopy velocity and influencing wave propagation through the vegetation field (Mei et al., 2011; Liu et al., 2015). The inclusion of the porosity effect can lead to possible reflection in wave-vegetation models ((Arnaud et al., 2017)), which contributes to wave-height reduction behind the vegetation. The porosity effect is relevant in dense mangrove fields (e.g., Fig. 1a), and also in porous brushwood groins made up of dense wooden sticks (Lucas, 2017). Vegetation density can be expressed by the frontal vegetation area per canopy volume (i.e., ) or by canopy porosity (n = 1-) (Nepf, 2011). However, with the same value of , porosity can be different, which may considerably influence wave propagation through vegetation fields.
In this paper, a new vegetation model is developed in the time-domain wave-modeling code SWASH (Zijlema et al., 2011). Besides the commonly considered drag force induced by vertical vegetation cylinders, the vegetation model also includes horizontal vegetation cylinders and the effects of porosity and inertia, providing a logical supplement to existing studies (see Table 1). SWASH is a general-purpose numerical tool for simulating unsteady, non-hydrostatic, and free-surface flow phenomena in coastal waters. It is chosen to implement a vegetation model because of its open-source nature and its efficiency in handling large 2D computational domains. In section 2, we introduce the vegetation model and implementation in SWASH. In addition, we present a theoretical model that handles wave damping by horizontal vegetation cylinders. In section 3, we focus on testing the implementation of drag force induced by both vertical and horizontal vegetation cylinders. Modeled wave dissipation by vertical vegetation cylinders is evaluated against measurements and existing models. The difference between vertical and horizontal vegetation cylinders in wave dissipation is identified by modeling. In section 4, we test the inclusion of the porosity effect against lab experimental data, and explore how porosity and inertia effects can affect wave transmission over vegetation patches. Finally, in section 5, we discuss the overall model performance, the influence of the included effects, and potential coastal-management implications and applications.
Section snippets
Governing equations
The SWASH model is a time-domain model for simulating non-hydrostatic, free-surface, and rotational flow. The governing equations are the shallow water equations, including a non-hydrostatic pressure term. We consider a two-dimensional domain that is bounded vertically by a free surface z = ζ(x,t) and a fixed bed z = −d(x) (see Fig. 1). Here t is time, x and z are the Cartesian coordinates, and z = 0 is the still-water level. The governing equations are:
Validation of drag force implementation of vertical and horizontal vegetation cylinders
As drag force contributes the bulk of wave dissipation, we first tested the drag force-oriented wave dissipation. Cases with vertical (sections 3.1 One-dimensional wave propagation over vertical vegetation on a flat bottom, 3.2 One-dimensional wave propagation over vertical vegetation on a slope, 3.3 Two-dimensional wave propagation over patchy vertical vegetation on a slope) and horizontal (section 3.4) vegetation cylinders were both explored. The tests with vertical cylinders included
Porosity and inertia effects in a solitary wave
In this section, we focus on validating the implementation of the porosity effect in the SWASH model. The solitary wave propagation in the physical model of Iimura and Tanaka (2012) is reproduced (Fig. 10d). The modeled maximum water level was compared against their measurement. In their physical model, the tested vegetation was always emergent. Three different stem densities were included: 462, 1283 and 11,547 stems/m2 in cases 1 to 3, respectively. As the tested was 0.005 m, spatial
Discussion and conclusions
In the present study, we represent a new wave-vegetation model implemented in SWASH. The implementation of the drag force (induced by vertical cylinders) is validated against established analytical and numerical models (Cao et al., 2015; Méndez and Losada, 2004) and laboratory measurements (Wu et al., 2011). It has been shown that the modeled wave reduction is in good agreement with existing models and measurements. Even in the submerged-vegetation condition, wave dissipation is reproduced well
Acknowledgements
We thank the two anonymous reviewers for their constructive comments. The new vegetation model described in this paper is available in the SWASH open source code (http://swash.sourceforge.net). The authors gratefully acknowledge financial support of the National Natural Science Foundation of China (No. 51609269), the Joint Research Project NSFC (No. 51761135022) – NWO (No. ALWSD.2016.026) – EPSRC (No. EP/R024537/1): Sustainable Deltas and NSFC grants (No. 51520105014 and No. 41771095). We thank
References (60)
- et al.
Wave propagation through dense vertical cylinder arrays: interference process and specific surface effects on damping
Appl. Ocean Res.
(2017) - et al.
Laboratory and numerical studies of wave damping by emergent and near-emergent wetland vegetation
Coast. Eng.
(2009) - et al.
How ecological engineering can serve in coastal protection
Ecol. Eng.
(2011) - et al.
Identifying knowledge gaps hampering application of intertidal habitats in coastal protection: opportunities & steps to take
Coast. Eng.
(2014) - et al.
On the one-dimensional steady and unsteady porous flow equations
Coast. Eng.
(1995) - et al.
Numerical modeling of vegetation-induced dissipation using an extended mild-slope equation
Ocean Eng.
(2015) - et al.
Deriving vegetation drag coefficients in combined wave-current flows by calibration and direct measurement methods
Adv. Water Resour.
(2018) - et al.
Numerical simulations of wave propagation over a vegetated platform
Coast. Eng.
(2016) - et al.
Sustainable hydraulic engineering through building with nature
J. Hydro-Environ. Res.
(2015) - et al.
Currents and sediment transport in mangrove forests
Estuar. Coast Shelf Sci.
(1997)
Laboratory study on wave dissipation by vegetation in combined current-wave flow
Coast. Eng.
Interaction of solitary waves with emergent, rigid vegetation
Ocean Eng.
Numerical simulation estimating effects of tree density distribution in coastal forest on tsunami mitigation
Ocean Eng.
Investigations on the porous media equations and resistance coefficients for coastal structures
Coast. Eng.
Wave damping over artificial Posidonia oceanica meadow: a large-scale experimental study
Coast. Eng.
Periodic water waves through an aquatic forest
Coast. Eng.
A new formulation for vegetation-induced damping under combined waves and currents
Coast. Eng.
Wave-induced dynamics of flexible blades
J. Fluids Struct.
Numerical study of turbulence and wave damping induced by vegetation canopies
Coast. Eng.
Solitary wave attenuation by vegetation patches
Adv. Water Resour.
Tsunami wave interaction with mangrove forests: a 3-D numerical approach
Coast. Eng.
A coupled model of submerged vegetation under oscillatory flow using Navier-Stokes equations
Coast. Eng.
An empirical model to estimate the propagation of random breaking and nonbreaking waves over vegetation fields
Coast. Eng.
Wave transformation over salt marshes: a field and numerical modelling study from north Norfolk, England
Estuar. Coast Shelf Sci.
Flow over and through biota
Non-hydrostatic modelling of infragravity waves under laboratory conditions
Coast. Eng.
Effects of emergent vegetation on lateral diffusion in wetlands
Water Res.
Efficient and robust wave overtopping estimation for impermeable coastal structures in shallow foreshores using SWASH
Coast. Eng.
Wave dissipation by vegetation with layer schematization in SWAN
Coast. Eng.
Numerical model for coastal wave propagation through mild slope zone in the presence of rigid vegetation
Coast. Eng.
Cited by (74)
Gaussian process regression approach for predicting wave attenuation through rigid vegetation
2024, Applied Ocean ResearchImproving wave height prediction accuracy with deep learning
2024, Ocean ModellingExperimental study on drag coefficient of flexible vegetation under non-breaking waves
2024, Ocean EngineeringHydrodynamic modeling study of nature-based hybrid coastal defense strategy applied in salt marsh restoration
2024, Estuarine, Coastal and Shelf ScienceExperimental investigation of wave attenuation by mangrove forests with submerged canopies
2023, Coastal Engineering