Wave action bottom reflection source term implementation

Highlights

• Analytical bottom reflection source term (ST) was implemented to WaveWatchIII.

• The ST derived accounts for the wave action time derivative.

• A new directional scheme was developed and implemented.

• The ST accounts for the wave bottom reflection in a large scale numerical model.

• Numerical investigations show good agreement to the test cases selected.

Abstract

Wave bottom reflection induced by a changing bathymetry has a major impact on the near-shore environment, where harbors and other infrastructure are based. However, this reflection is only partially accounted for in large scale numerical wave prediction models. The present work implements the analytical bottom reflection source term derived in Yevnin and Toledo (2018) to WAVEWATCH III, a numerical prediction model based on the wave action equation. The new source terms reflection coefficient magnitude, reflection direction and directional spread are shown to produce good results with respect to analytical and numerical solutions, and are tested in a real-world example. The new term allows to better simulate the near shore environment, especially near islands, reefs, semi-enclosed basins or natural beaches with long mild slopes, where the directional changes can cause previously unaccounted for effects.

Introduction

Wave reflection from the sea bottom, coastline or submerged obstacles can have a profound impact on wave behavior in coastal waters and even in deep water (see for example O’Reilly et al., 1999). In the near-shore area it can, for example, cause harbor agitation as waves reflect from nearby bathymetry in different directions, penetrating bays and harbors, which may cause serious damage to ships and infrastructure (see Mei et al., 2005, Tobias and Stiassnie, 2011 for harbor agitation and shelf edge respectively). It can also cause unaccounted for wavefields around islands, as waves could be reflected and even trapped by changing bathymetry (Liu et al., 2004). In deep water it can cause nonlinear triad interactions, leading to infragravity waves and secondary peaks, which are not yet accounted for in numerical forecasting models. (for forward propagating waves, see Vrećica and Toledo, 2016, Vrećica et al., 2019).

Wave reflection (and refraction) is caused when waves propagate over a changing bathymetry, where part of the wave energy is transmitted forward and some of the energy is reflected off the sea bottom (Meyer, 1979). The magnitude of reflection is affected by the bottom depth and slope, and by the wave period, height and direction, all of which can change locally. Investigations of such reflection have shown it could have up to 40% wave action reflected at steep cliffs (O’Reilly et al., 1999), but more commonly around 5%–10% reflection over natural beaches (Tatavarti et al., 1988, Elgar et al., 1994). Few studies have been done on the directional effect of such reflections, due to the difficulty of measuring wave reflection and separating it from general wave propagation in the chaotic coastal environment. The effects of these reflections can be sensed mostly by examining directional spread (Kuik et al., 1988) differences of models and measurement devices, as suggested in Ardhuin and Roland (2012).

Two wave reflection source terms were introduced into wave forecasting models, such as WAVEWATCH III (Tolman, 1991) and SWAN (Booij et al., 1999). The first, presented in Ardhuin and Magne (2007) and based on Ardhuin and Herbers (2002) and Magne et al. (2005), was designed for sub-grid topography undulations. The second aims to recover the reflected wave action of shoaling waves. This source term was presented in Ardhuin and Roland (2012), where wave action reflection is calculated at the coastline. This approach is somewhat limited, for example in cases where submerged obstacles are present, or when the topography is close to the water surface but does not reach land. Another limiting factor is the reflection coefficient, that has to be calibrated to achieve good results. This is very difficult since, as stated above, measuring wave reflection directly is complicated.

The source term for coastal reflection used in WAVEWATCH III (WW3) was derived in Ardhuin and Roland (2012) (here referred to as REF1) as:

$$N_{ref,unst}\left(f,\theta \right)=\underset{0}{\overset{2\pi }{\int }}{R}^2\left(f,\theta ,{\theta }^{\prime }\right)N\left(f,{\theta }^{\prime }\right)d{\theta }^{\prime },$$

where $N_{ref,unst}$ is the source term for unstructured grid; $R^2$ is the energy reflection coefficient; $N$ is the wave action directional spectrum; $f$ is the wave frequency; ${\theta }^{\prime }$ is the incident direction; and $\theta$ is the reflection direction. The reflection coefficient has been based on the Miche number — empirically related to the bottom slope, the deep water wave height and the wave frequency (Miche, 1951). This source term is currently used by several wave forecasting models as an optional term to account for wave reflection. The reflection can be locally calculated automatically through the Miche number, although it was suggested that this could be sensitive to bathymetric grid coarseness and the resolution of the shoreface slope. For this reason, it was suggested by the original authors that a manually set, model-wide, constant reflection coefficient be used if the shoreface slope cannot be otherwise simulated. Furthermore, this source term relies only on land-based reflection, and the coastline shape and direction (straight, oblique or sharp angle). This means shallow water without adjacent land in the model does not produce any reflected waves.

In Yevnin and Toledo (2018) a steady-state analytical term for bottom reflection was introduced as:

$$\frac{\partial }{\partial \xi }\left(\frac{k_{\xi }}{k}{C}_g{N}_{ref}\right)=|-i{\left(\frac{\partial k_{\xi }∕\partial \xi }{2k_{\xi }}\right)}^2\frac{C_g{N}_{in}}{k}{e}^{4i\int k_{\xi }d\xi }|,$$

where $N_{ref}$ is the reflected wave action directional spectrum; $i$ is the imaginary unit; $\xi$ is the bottom gradient direction; $k$ and $k_{\xi }$ are the wavenumber and wavenumber in the $\xi$-direction respectively; $C_g$ is the group velocity; and $N_{in}$ is the incident wave action directional spectrum. It was based on a perturbation analysis and a WKB approximation of the mild-slope equation (see Berkhoff, 1972, Booij, 1983), and assumed no changes in time (steady state). It accounts for varying bottom slopes, wave periods and oblique incident angles. The source term was compared with the REF1 source term, and showed more accurate results, reaching even into the infragravity wave range, under mild-slope assumptions (for further details see Yevnin and Toledo, 2018).

In this paper an implementation of the analytical source term to WW3 is presented, followed by testing and verification. The paper is structured as follows: In Section 2 the source term is extended to account for the time derivative of the wave action, thus more accurate and better fitting for implementation; In Section 3 a new directional scheme for the local coordinate system used in the source term is formulated and explained. The reflection source term implementation in WW3 is tested in Section 4. Finally, in Section 5 the work is summarized and discussed.