Wave-tide interaction modulates nearshore wave height

Coastal flooding from an extreme sea level event is driven by the combination of storm tides (tides and surges) and waves (Lewis et al. 2013, 2011). This extreme water-level condition can overtop or breach a coastal defence, resulting in inundation (e.g. Brown et al. 2010). It is therefore crucial to understand how these flooding processes interact, and combine, to increase the hazard—especially in the coming century as coastal flood risk is expected to increase (e.g. Leonard et al. 2014; Arns et al. 2017).

Waves contribute directly to an extreme water-level through wave breaking and run-up process; for example, wave set-up at a local level has been calculated to further increase the water levels by up to 0.5–1.0 m in Liverpool Bay (Brown 2010). Wave overtopping of a coastal defence can lead to inundation (Wolf 2008; Phillips et al. 2017), but can also cause erosion (Wolf 2009), enabling waves to propagate further inshore and thus increasing flood risk: through increased wave overtopping and the possibility of flood defence failure (van der Meer et al. 2009). Flood risk is also influenced by the pre-storm beach morphology and defence fragility, both of which may be changed during a succession of events or seasonal variability in beach level. Therefore, the combination hazard of waves and extreme high waters is essential to resolve within flood risk understanding, especially as wave and tides are known to interact (Wolf 2009).

In order to operationally determine the extreme water-level hazard for a given scenario, hydrodynamic models at a national scale are often used to provide boundary conditions to local storm impact models, which are used to simulate the consequent wave run-up, wave overtopping, or overwash (Souza et al. 2013). Phase averaged spectral wave models and shallow-water equation models are typically used at the basin-scale. Higher fidelity (wave resolving and non-hydrostatic hydrodynamic) models are typically used within the surf zone to determine the hazard drivers impacting coastal defence schemes, with wave overtopping and inundation models used beyond the flood defence (e.g. Prime et al. 2015).

In many recently developed systems, unstructured modelling techniques have been applied in order to resolve many of the physical processes within an extreme flooding event from the basin-scale to local-scale and inundation/damage extent (e.g. Westerink et al. 2008; Bunya et al. 2010; Dietrich et al. 2011; Roland et al. 2012; Hope et al. 2013). However, compound flood risk and wave-tide interaction effects on coastal flooding in particular have often been understudied. For example, although large-scale numerical models simulate wave and storm tide (tide plus storm surge) processes, these are often run independently in operational forecast systems. In coastal flood systems design, there is a tendency to concentrate on the first order effect of depth variations on waves at the local scale, whilst neglecting potentially important broader scale effects that may be had on the wave field as a result of interactions between waves and current offshore (e.g. Ardhuin et al. 2017).

The analytical solution to the effect of currents on waves (in the absence of wave breaking processes) is given by Phillips (1977): \( \frac{A}{A_0}=\frac{c_0}{\sqrt{c\left(c+2U\right)}} \); where wave amplitude (A) and phase speed (C) are modified (to A₀ and c₀) due to an ambient current (U). The resultant change in water depth due to tidal elevation also changes wave refraction, which can modulate the nearshore wave climate (Hashemi et al. 2016). Therefore, in regions of strong tidal dynamics, wave-tide interaction is likely to significantly affect the far-field wave climate propagating towards the coast and thus coastal hazards.

The interaction of tides and other flood risk drivers (i.e. waves) is important to understand, particularly in regions of large tidal range where flooding coincides with the high tide (Pugh 1996; Muis et al. 2016). Analysis of global tidal data (FES2012; Carrere et al. 2012) indicates that ~ 10% of the world’s coastlines experience marco-tidal conditions (tidal range > 4 m, using the four major tidal constituents of M2, S2, K1 and O1; see Neill et al. 2018). Over 600 million people are estimated to live in low-lying coastal regions (McGranahan et al. 2007; Muis et al. 2016), and, as coastal bathymetry often enhances tidal currents (Lewis et al. 2015), the contribution of wave-tide interaction to combination flood hazard (and how this may change) is essential to resolve. For example, the UK is one such region with macro-tidal conditions (including the second largest tidal range in the world), where approximately 6 million properties are exposed to some degree of flood risk (Prime et al. 2015).

The interaction between the tide and surge is well-known (e.g. Horsburgh and Wilson 2007), and is included with extreme water-level estimates using joint probability methods (e.g. applying skew surge to the harmonically predicted tide). The co-occurrence of waves and extreme water levels is also included within rigorous flood risk assessments (e.g. Prime et al. 2016); however, the modulation of offshore wave height due to wave-tide interaction processes is typically omitted in flood risk scenarios. Hence, if waves are likely to be larger towards times of high water, when flood risk is higher, wave-tide interaction needs to be included within flood risk assessment (Wolf 2009).

The COAWST modelling system is used here to simulate the dynamic interaction between waves and tides. The COAWST model has been successfully implemented in a number of studies of the Irish Sea region (e.g. Lewis et al. 2014), and is similar to previous studies of wave-tide interaction in the Irish Sea (Brown et al. 2010; Brown et al. 2011). To determine the effect of tidal dynamics on wave propagation and generation in the Irish Sea, the wave model was run both coupled and uncoupled. The difference in simulated wave height between the coupled and uncoupled model runs was used to determine the modulation of wave height due to the tide.

The COAWST system comprises of the ocean model Regional Ocean Modelling System (ROMS), the atmospheric model WRF and the Simulating Waves Nearshore (SWAN) wave model—which is described in Warner et al. (2008a, b) and Hashemi et al. (2015). Data exchange between these modules is conducted with the Model Coupling Toolkit (Warner et al. 2008b; Warner et al. 2010). Two-way processes coupling in COAWST includes wave refraction by currents, bottom friction due to combined currents and waves, enhanced wind drag due to waves, and 3D interactions of stokes drift, radiation stress and Doppler velocity. Therefore, wave-tide interaction can be simulated to investigate the effect of the tide on the modulation of wave height.

Three model spatial resolutions were used to investigate this effect to our results; defined here as Coarse (1/60°), Medium (1/120°), Fine (1/240°), which is consistent with previous studies (Lewis et al. 2015). The “Medium grid” was used to investigate potential changes from mean sea-level rise and the overall effect of tides on waves. Two months were simulated (January–February 2014) as both extreme (wave heights and wave periods in excess of 8 m and 11 s, respectively) and quiescent conditions occurred. Model computed wave and tide results from this 2-month period were output at 1 h frequency, with data exchange between the wave and tide model every 600 s (an unpublished sensitivity test showed no significant difference to results with changes to data exchange below this frequency).

2.2 ROMS hydrodynamic model

The ROMS is used to simulate tidal dynamics in the Irish Sea. The ROMS modelling system uses a finite-difference approximation of the 3D Reynolds-Averaged Navier-Stokes (RANS) equations, with hydrostatic and Boussinesq assumptions, with a horizontal curvilinear Arakawa C grid and terrain-following vertical coordinate system (the sigma coordinate system) of 10 depth layers evenly spaced throughout the water column (see Lewis et al. 2015). This ROMS model has been successfully applied to previous Irish Sea tidal modelling studies (Lewis et al. 2015, 2017). Further details of the ROMS model, including the discretised full equations and model verification details, are found in a number of publications (e.g. Shchepetkin and McWilliams 2005; Haidvogel et al. 2008), and so are not described further here (see also www.myroms.org).

Digitised Admiralty data (at 200 m resolution; http://digimap.edina.ac.uk) was interpolated to the ROMS computational grid using a nearest neighbour approach, and corrected to mean sea-level. The domain and the bathymetry of the Irish Sea tide model are shown in Fig. 1. The geographic scale of inter-tidal regions is relatively small in relation to model resolution and extent (Lewis et al. 2015); therefore, a minimum water depth of 10 m was applied to the bathymetry data (i.e. no wetting and drying in the model), as has been previously applied to this region (e.g. Lewis et al. 2015). The open boundary of the tidal model was forced with ten tidal constituents (M2, S2, N2, K2, K1, O1, P1, Q1, Mf and Mm) from the Finite Element Solution and data assimilated global tide product (FES2012) by Carrere et al. (2012), which has shown to be accurate in this region (Lewis et al. 2015).

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A constant drag coefficient (C_D) of 0.003 was assumed within the quadratic friction model parameterisation, consistent with previous studies (e.g. Lewis et al. 2014). The interaction between waves and bed shear stress is parameterised using the SSW-BBL option in COAWST (Warner et al. 2010). In the SSW-BBL routine, the artificial bed roughness (Z0) from wave-current interaction is based on median sediment grain size (D50) of 3 mm—consistent with previous studies (e.g. Lewis et al. 2014).

The turbulence closure Generic Length Scale (GLS) model was tuned to the κ-ε turbulence model, including wave breaking and wave effects on current (WEC) vortex-force parameterisation (Kumar et al. 2012). Here, we use the WEC_MELLOR (Warner et al. 2010) parameterisation of radiation stress terms for waves on currents—as previously applied in the Irish Sea (Lewis et al. 2014). Therefore, conservative and non-conservative wave force effects on the simulated mean flow-field of ROMS are included within the COAWST model.

2.3 Model validation

The COAWST Irish Sea model was validated for both tides and waves (Table 1) and the location of validation data shown in Fig. 1. Eleven tide gauges (diamonds in Fig. 1) were used to validate tidal elevation, and example time-series are shown in Fig. 2. To validate the model for tidal currents, 140 sites were used (9 sites of depth averaged currents and 131 depth specific M2 tidal current sites) using data, quality controlled and processed into principle semi-diurnal lunar constituent, M2, tidal current data by the British Oceanographic Data Centre (www.bodc.ac.uk). Validation showed accurate simulation of tidal dynamics (~ 5 and 10% Normalised Root Mean Squared Error for M2 elevation and currents, respectively, and an overall R² value of 99%). There were no major differences in validation statistics (i.e. Normalised Root Mean Squared Error or Rsq skill) between model spatial resolutions; see Table 1.

Table 1

Model validation for three spatial resolutions using Root Mean Squared Error (RMSE) with (Normalised-RMSE in brackets as %). Data from 11 tide gauges (amplitude (amp) and phase (pha) of M2 and S2 tidal constituents), 9 depth-averaged M2 tidal ellipse data (major axis (Cmax), minor axis (Cmin), Inclination (Inc) and phase), 131 M2 tidal current observations at specific depths (U and V amplitude and phase), wave height (Hs) and period (T) for three wave buoy records and eight satellite tracks

		Coarse (1/60°)	Medium (1/120°)	Fine (1/240°)
		RMSE (NRMSE)	RMSE (NRMSE)	RMSE (NRMSE)
Tidal elevation	M2 amp	0.13 m (5%)	0.12 m (5%)	0.11 m (4%)
	M2 pha	6°	6°	4°
	S2 amp	0.08 m (9%)	0.08 m (9%)	0.08 m (9%)
	S2 pha	14°	14°	9°
Depth-mean M2 tidal ellipse	Cmax	0.08 m/s (11%)	0.07 m/s (10%)	0.07 m/s (10%)
	Cmin	0.02 m/s (8%)	0.02 m/s (8%)	0.02 m/s (8%)
	Inc	6°	5°	5°
	Pha	8°	7°	7°
M2 currents at specific depths	M2 Uamp	0.10 m/s (9%)	0.11 m/s (10%)	0.11 m/s (10%)
	M2 Upha	4°	4°	4°
	M2 Vamp	0.09 m/s (9%)	0.09 m/s (10%)	0.09 m/s (10%)
	M2 Vpha	6°	6°	6°
Wave buoys	Hs	0.47 m (9%)	0.58 m (11%)	0.50 m (10%)
	T	1.4 s (22%)	1.8 s (32%)	1.8 s (32%)
Satellite	Hs	0.50 m (34%)	0.51 m (36%)	0.50 m (34%)

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The ROMS, sigma coordinate, 3D current fields were interpolated to validation data using a nearest neighbour approach. All tidal current validation stations were in water depths greater than 10 m. Further details of this data can be found in Lewis et al. (2015) and only briefly described here: Water depth of the current measurements varied between 143 and 3 m (mean of 58 m), which gave a wide range of 3D current validation data between the near seabed (max sensor depth at ~ 99% of water depth) to near surface (minimum sensor depth ~ 4% of water depth) and mid water column (mean sensor height at ~ 43% of water depth); as well as a wide range of current speeds (major axis of the principle semi-diurnal lunar constituent (M2) ranged between 0.03 and 1.10 m/s, with an average of 0.54 m/s).

To spatially and temporally validate the accuracy of the simulated wave climate, three wave buoys and eight satellite tracks were used (see Fig. 1). A 56-day record (3 January 2014 to 28 February 2014) from the Met Office “Aberporth” (Ab) wave buoy (52.37°N 4.69°W in 46 m water depth), a 7-day record (1 January 2014 to 11 January 2014) from the Met Erin “M2” station (53.48°N 5.43°W in 85 m water depth) and a 45-day record (3 October 2014 to 16 November 2014) from the Bangor University “Seacams” (SC) wave buoy (53.22°N 4.72°W in 46 m water depth) were used as the validation wave buoys. Wave height data from the eight satellite tracks were made available through the e-surge product (www.storm-surge.info/) and compared to the simulated wave height data at nearest model cell. The location of the satellite tracks is shown in Fig. 1, and the dates listed in Table 2.

Table 2

Satellite track data for wave height spatial validation; see Figs. 1 and 4

Track (see Fig. 1 for position)	T1	T2	T3	T4	T5	T6	T7	T8
Date (dd/mm/yy) and time range	1/1/14 10:01 to 10:52	3/1/14 17:01 to 17:33	4/1/14 10:01 to 10:04	7/1/14 17:01 to 17:07	9/1/14 22:01 to 22:24	10/1/14 16:01 to 16:19	12/2/14 09:02 to 09:29	13/2/14 02:02

Overall, significant wave height (Hs) and mean wave period were simulated with accuracy (RMSE of ~ 0.5 m and < 2 s, respectively) and skill (e.g. R² of ~ 85% overall for Hs). Validation statistics were broadly similar for all three model spatial resolutions; as summarised in Table 1. Validation statistics (NRMSE and Rsq) were also broadly similar between the coupled (COAWST) and the uncoupled (SWAN-only) model at medium resolution (1/120° grid); an accuracy difference of 0.01 m (< 1% NRMSE) in wave height and 0.2 s (< 2% NRMSE) in wave period, with an Rsq difference of 2% for both these parameters: as shown in Table 3 in the appendix. The comparison of wave height simulated with the coupled and uncoupled model is shown in the more detailed analysis of wave buoys and satellite tracks in Figs. 3 and 4, respectively.

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The ability of the coupled model to simulate the effect of tides on waves is further visually illustrated in Fig. 5. A short example of wave height modulating at the semi-diurnal period (~ 12.42 h), observed at the Aberporth wave buoy, is shown in Fig. 5a. A one-dimensional wave spectra comparison at the Aberporth buoy site (using FFT analysis in Matlab with a Nyquist frequency of 7200 s) found the coupled model recreates the observed 12.42 h modulation of wave height in the 2-month record (January–February 2014), whilst the wave-only uncoupled model does not, as shown in Fig. 5b. We therefore have confidence in the accuracy of the wave-tide coupled (COAWST) Irish Sea model to simulate wave-tide interactions effects (see also Table 1).

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The effect of wave-tide interactions on high water wave height was explored using a wave-tide coupled model, for a 2-month period (January and February in 2014) observed to be the stormiest period of weather within 20 years (Huntingford et al. 2014; Kendon and McCarthy 2015) when a number of high-energy wave events resulted in coastal flooding (Sibley et al. 2015). The simulated tidal modulation of wave height matches previous studies of the Irish Sea, where tides modulated wave heights by ~ 10% in some areas (Brown et al. 2011; Hashemi et al. 2015).

In our study, wave-tide interaction increased high water wave height by up to 20% for regions of the coastline where modulation of waves coincided with local high water (Fig. 7). As overtopping hazard increases exponentially with wave height (Pullen et al. 2007), such an increase is also likely to exponentially increase flood risk (see Stockdon et al. 2006). Therefore, for these regions of increased wave height at high water, boundary condition uncertainty needs to be propagated into flood hazard impact models, and joint probability modelling scenarios of flood risk required (e.g. Prime et al. 2016). For example, the modulation of wave height in a tidal signal could be propagated through overtopping models (e.g. EUROTOP; Pullen et al. 2007), and multiple flooding drivers (e.g. erosion effects and tidal variations to wave height), then cascaded through to inundation extent (e.g. within LISFLOOD modelling frameworks; see Gallien et al. 2014).

Increased wave action on coastal defences (altering coastal defence fragility and wave overtopping rates), as well as wave erosion and increased wave run-up (e.g. Wolf 2009) is likely to further increase the combined flood risk, when considering the tidal modulation of nearshore wave height shown in this study; however, the accurate assessment to the combination hazard of coastal flooding is likely to be a complex task, requiring multiple input parameters (beach morphology, coastal defence structures, river discharge, water-level and wave climate). For example, the spatial scales of the modelling presented in this study are too coarse for drawing many if any conclusions in the nearshore portions of the domain (i.e. our study does not resolve processes in the surf zone). However, the results shown here could be propagated through higher fidelity coastal zone modelling techniques to resolve the likely impact of wave-tide interaction to coastal flood risk. One potential solution could be an unstructured modelling approach to resolve much of the surf zone processes omitted within the study (e.g. Bunya et al. 2010; Dietrich et al. 2011; Roland et al. 2012; Hope et al. 2013); however, the computational burden of simulating combined hazard events at regional-scale (i.e. multi-month morphodynamic, wave-tide coupled, fluvial and oceanic flood risk driver simulations) would be exceedingly challenging (especially at regional-scale, with multi-ensemble runs such as typical within flood forecasting).

The spatial variability of wave height affected by wave-tide interaction processes (Fig. 9) indicates the process behind higher waves at high tide for some regions (Fig. 7): a large effect of the tide on wave height was found around features where tidal currents are accelerated, such as the headlands of Wales (Pembrokeshire in the south and the Llyn in the north; see Fig. 9) and as predicted by Phillips (1977). Further, the tidal current field will affect refraction of waves due to Doppler shift (such as around the large-scale headland features in North and South Wales), resulting in the modulation of wave propagation into the bay features behind these headlands (e.g. Cardigan Bay) and for large areas of the Irish Sea (waves predominantly propagate northwards from the Atlantic exposed south).

The timing of the tidal modulation to wave height (e.g. Fig. 5) coincides as an increase in wave height around high water for some regions (Fig. 7), with clear flood risk implications. However, wave height was also found to increase around low water in other regions, especially for regions of strong tidal currents, and this may result in changes to erosion and sediment transport pathways due to enhanced bed shear stress from the combined effect of waves and currents (Hashemi et al. 2015). Future work should therefore investigate potential changes to sediment transport pathways from wave-tide interaction and likely sea-level rise. Moreover, surges will also change water depth (and currents) and river input could be very important for communities within estuaries; therefore, wave-tide-surge-river interaction could also be the focus of future work for accurate representation of extreme flood hazard conditions (e.g. Maskell et al. 2013).

Clear differences in the simulated wave height at high tide were found using three model resolutions, although the pattern was broadly similar (Fig. 10). As tidal currents are accelerated by bathymetric features, coupled wave-tide models need to include these fine-scale spatial features. Hence, model spatial resolution is important for accurate simulation of tidal current fields around complex coastlines, whilst spatial variability of tidal elevation is small (see Fig. 11). Previous studies have shown sub-kilometre-scale resolution is needed to accurately resolve tidal currents (Lewis et al. 2015), and therefore a similar resolution should be sought after in wave-tide coupled modelling at shelf scales—such as applying an unstructured grid approach (e.g. Roland et al. 2012). Furthermore, differences in the simulated tidal dynamics were clear between the three model resolutions, and yet their validation was broadly similar (Table 1). Indeed, in some regions the difference in simulated wave height between the coupled and uncoupled models was of the same order of magnitude as the validation. Therefore, future work should investigate improved methods of validation—beyond our visual comparison (Fig. 5) and traditional methods (e.g. nearshore wave climate observations and improved validation statistics, such as wavelet analysis; see Palmer et al. 2015).

The effect of sea-level rise was shown to increase the simulated wave-tide interaction processes for three scenarios: 0.44, 0.74 and 2.00 m mean sea-level rise. The increased water depth, and subsequent reduced tidal friction within our model, significantly changes both tidal amplitudes (Fig. 12) and tidal currents (Fig. 13) in the Irish Sea. The enhanced tidal dynamics from sea-level rise broadly match previous modelling studies of Pickering et al. (2012, 2017); however, there is uncertainty in simulating tidal changes due to sea-level rise—because of assumptions in coastline adjustment and tidal friction parameterisation in the model (i.e. no changes to tidal boundary conditions, no wetting and drying, and no morphodynamical changes were included). Nevertheless, we have shown wave-tide interaction to be sensitive to changes in tidal dynamics—as sea-level rise-induced changes to tidal dynamics altered high water wave heights for all three scenarios throughout the computational domain (Fig. 11). Therefore, the tidal modulation of wave height may increase in the future due to sea-level rise-induced changes to tidal dynamics, which has implications globally, as the combined hazard of coastal flooding due to large waves at high tide may increase.

Dynamically coupled wave-tide modelling methods allow wave-tide interaction, and potential future changes to this process, to be investigated. The effect of tides in modulating nearshore wave height is investigated in the Irish Sea, with some regions experiencing a 20% increase in high water wave height due to tidal currents and elevation affecting the wave field propagating into a coastal region. Such potential changes in the nearshore conditions will impact on flood hazard, and this uncertainty/error could propagate through to flood risk assessment. Therefore, wave-tide interaction has clear implications to the combined hazard of flooding from extreme sea-level and waves.

Simulated wave-tide interaction was shown to be sensitive to spatial model resolution because tidal currents are enhanced by bathymetric features requiring higher spatial resolution (e.g. an unstructured grid modelling approach) to simulate the current field accurately. Therefore, coupled models must therefore have rigorous validation strategies to ensure accuracy, and sub-kilometre-scale grid resolution appears necessary to capture tidal flow around bathymetric features (such as headlands). Moreover, wave-tide interaction appears sensitive to potential future changes to tidal dynamics due to sea-level rise, with an increase in high water wave height. Future flood risk may increase in some regions as a result of the combined hazard of large waves occurring at times of high water—which has implications to flood risk mitigation strategy.