A comprehensive study of the tides around the Welsh coastal waters Estuarine, Coastal and Shelf Science

A computational model has been used to explore characteristics of the barotropic tide around the Welsh coast in detail for the first time. Proper understanding of tidal characteristics is vital for the sustainable use of marine resources; particularly for industries such as marine energy extraction, aggregate mining, aquaculture, as well as regulators and agencies with responsibilities for the resource management and public safety. In shallow water areas, the influence of bathymetry and energy dissipation leads to the generation of higher harmonics that cause complex tidal phenomena. The Celtic and Irish seas, which enclose the Welsh coast (UK), are heavily industri-alised shallow water seas with macro-to mega-tidal semi-diurnal tides. It is shown that tidal distortion is sig- nificant in the Bristol Channel (S. Wales) and in the large shallow estuaries of the N. Wales coast; for much of the west coast this is only significant in localised areas around headlands and islands. Tidal dominance switches from flood dominant in the south and north to ebb dominant on the west coast. Highly complex patterns of vorticity in the tidal residual flow are noted. All these factors mean that careful siting of industry and coastal management interventions is required to avoid disruption of the natural system.


Introduction
One of the major challenges for coastal engineers and coastal managers is addressing the risk of coastal flooding. Less visible but of similar importance are the changes in seabed morphology due to sediment transport by waves and tides. Such changes may alter both the flood risk but also the sustainability of ports and harbours, as well as threatening the ecology of environmentally important coastal sites, (Environment Agency, 2021). An understanding of the tidal variations in sea level and currents is an important element in planning coastal defence, shoreline management strategies and siting of marine energy extraction. Potential exists for both tidal stream energy projects, (Bryden and Couch, 2006), that extract energy from the tidal flow, and tidal range projects that generate energy from the variation in potential energy provided by high tidal range, . Further, it has been suggested that large projects such as the Severn Barrage would modify the flows to an extent that the performance of distant tidal energy schemes might be compromised, Willis et al. (2010). The primary strategy for resolving potential conflicts across the range of interests in using the seas is Marine Spatial Planning, a recent example of this within the region of interest is the Wales National Marine Plan (Welsh Government, 2019). Marine and shoreline plans require consistency and one of the main elements linking the two are the tides.
The tide generating forces arise from the differential gravitational attraction of the Moon and Sun experienced over the surface of the Earth. The oceans respond dynamically to these forces and interact with the irregular seabed and continents to create a complex and changing pattern of interference, or amphidromic systems. As a stratified fluid the ocean can support both external (barotropic) and internal (baroclinic) tides. With the exception of some highly stratified situations, the barotropic tides contribute most to the rise and fall of the sea surface and provide a very good approximation to the measured tidal level variations, (Cummins and Oey, 1997), and are the focus of this paper. Navigation, aquaculture and flood defence interests all require information on the daily variation of tide levels and currents. However, longer term tidal influences, arising from the cumulative effect of small asymmetries in the tidal oscillation, can have significant impacts on pollutant transport and sediment movements, (e.g. Prandle, 1984;McCave, 1970).
There is a vast literature on deep ocean tides and their propagation into shallower shelf seas. Tides around continents often have amplified magnitudes in relation to deep ocean tides. Fluid bodies forced by oscillations close to their natural frequency can have a large amplitude response; a process called resonance (Pugh, 1981), which occurs when the continental shelf is ¼ wavelength wide. A similar process can occur at a constriction of the sea, such as the English Channel, the Bristol Channel or the northern end of the Irish Sea. An exactly analogous situation exists at smaller scale when long waves encounter the entrance of an inlet or harbour. In the case of estuaries the amplification of the tidal oscillation is explained by shoaling caused by the decrease in water depth, and funnelling due to geographic constriction of the wave front. In the idealised case of no frictional losses, conservation of the tidal energy flux along a channel implies amplification of the tide progressing up a channel according to Green's law, (Jay 1991). Resonance, per se, requires reflection of the incoming wave and constructive interference between incoming and reflected wave. In harbours reflection is caused by the hard walls and breakwaters; in engineered estuaries it may be a barrier or weir that restricts the propagation of the tidal wave; and in natural estuaries reflection may occur due to a rapid change in bed level at the transition from flood plain to tidal river. Vigorous tidal motions in shelf seas arise from resonance with ocean tides and from local amplification due to the seabed configuration rather than direct action from the tide generating forces, (Pingree and Griffiths, 1987;Kang et al., 1998). This amplification can be dramatic in funnel shaped and resonant channels, as in the case of the Bay of Fundy (Canada) and the Bristol Channel (UK) where the maximum spring tidal range can exceed 14 m. The tides around the Welsh coast are highly amplified, in some locations being classified as mega-tidal; where the tidal range is in excess of 8 m.
Tidal amplification is modified by frictional momentum dissipation and nonlinear interactions that distort the sinusoidal character of the deep ocean tides and generate shallow water harmonics: overtides which have frequencies that are multiples of those of the basic astronomical constituents, for instance M 4 that has a frequency twice that of M 2 ; and compound tides which arise through interactions of two different harmonics, for instance MS 4 that has a frequency equal to the sum of those of M 2 and S 2 , (see e.g. Aubrey and Speer, 1985;Friedrichs and Aubrey, 1988). An important consequence of tidal wave distortion is the asymmetry of tidal motion in shallow water: the periods of tidal rise and tidal fall may not be equal; the magnitude and duration of flood and ebb tidal flows may be unequal, (Gallo and Vinzon, 2005). This asymmetry can lead to preferential transport of sediments by the tidal oscillation either into or out of an estuary. Understanding tidal asymmetry is important from the marine renewable perspective too, as high tidal asymmetry can adversely affect tidal power extraction due to the high dominance of either an ebb or flood flow, (Neill et al., 2014;Ward et al., 2018). Further, Neill et al. (2009) demonstrated that tidal energy extraction can modify the hydrodynamics of a region and in areas of high non-linearity may have significant consequences for the dynamics of sediments due to the generation of residual flows.
Asymmetry in the tidal oscillation leads to a net drift or tidal residual current. Sediment transport is widely considered to obey a power law with respect to the tidal velocity, (e.g. Lavelle et al., 1984;Van Rijn, 1993). The quasi-oscillatory nature of tidal currents suggests net sediment movement due to tidal currents might be in the direction of the residual current direction. This perspective is supported by the correlation of tidal residual currents with long term sediment movements, (see e.g. McCave, 1970;De Swart and Zimmerman, 2009;Moore et al., 2009).
The origin of tidal asymmetries is through non-linear dynamics such as advection and bed shear stress. Although tidal forcing at the mouth of an estuary or embayment is often at diurnal or semi-diurnal frequencies, the nonlinear terms in the momentum equations almost always produce significant higher harmonics in shallow water (Aubrey and Speer, 1985). Ianniello (1979) showed that while first-order tidal propagation is relatively insensitive to small bathymetric variations the associated nonlinearities result in the generation of significant higher harmonic and residual components, with frictional effects being pronounced in strongly convergent channels, irrespective of depth, (Friedrichs and Aubrey, 1994). Tidal asymmetries are very often associated with the quarter-diurnal tidal component, M 4 , which tends to be more pronounced in locations where the coastal zone is shallow and the tidal amplitude is large, (Song et al., 2011). This may be understood in a simple way from the trigonometric relationship in which the product of two harmonics of frequencies ω 1 and ω 2 and amplitudes U 1 and U 2 results in constituents at their sum and difference frequencies. Thus, terms involving the product of the M 2 harmonic generate both M 4 and Z 0 harmonics; an overtide and a residual (constant). Similarly, M 2 and S 2 will generate ω 1 + ω 2 = ω MS2 and ω 2 -ω 1 = ω MSf , that is, quarter-diurnal and fortnightly periods. In fact, as discussed in Prandle (2009), the advective terms act in this manner but the quadratic friction term generates odd harmonics (i.e. M 6 and M 10 from M 2 ) as well as M 4 . These harmonics play a crucial part in the seabed sediment dynamics and in the net transport of pollutants (Pingree and Griffiths, 1979). They are also sensitive to the details of the local bathymetry and can be difficult to predict with traditional methods, (Adcock and Draper, 2014). The practical application of such theory is illustrated by the study of Shapiro (2011) who showed that tidal power extraction may affect the residual circulation as far as 100 km away from its location.
Here, a distinction is made between the residual (of either elevation or current) obtained by an harmonic analysis of tide gauge records and the residual obtained from a computational simulation of tidal flow. The former will likely contain contributions from non-tidal sources such as waves, surface pressure and surface winds. The latter arise solely due to asymmetries in the tidal flows caused by nonlinear interactions and bathymetric configuration, and are termed tidal residuals henceforward.
Modelling studies by Prandle (1978Prandle ( , 1984 showed typical tidal residual currents of 1-3 cm/s over the continental shelf around the UK. Tidal residual currents are usually much smaller than flood or ebb currents but their significance to sediment movement should be viewed from the perspective of their action over many months or years. Maps of residual currents often reveal the presence of closed circulation cells or 'gyres' that can act to trap mobile sediment, (e.g. Takasugi et al., 1994). The existence of gyres can be understood as a transfer of vorticity from the fluctuating tidal motion to the mean (residual) field, (Robinson, 1983). The most important vorticity generation mechanism is the squeezing and stretching of the water column over the sea topography (Zimmerman, 1981), while the torque from the bottom friction force may also be significant, (Robinson, 1983;Ridderinkhof, 1989). The vorticity provides a quantitative measure of the strength of gyres in the residual flow. Zimmerman (1981) argued from theoretical considerations that while bottom morphology influenced the residual currents there were also feedbacks from the residual currents in altering the bottom morphology. This is important in areas with morphological features composed of mobile sediment (e.g. sandbanks and sand waves), in which residual currents will shape the sandbanks and at the same time the sandbanks will shape the residual current flow. An explanation for the growth and maintenance of sandbanks in terms of residual currents was provided by Huthnance (1982) and subsequently expanded to explain the movement of sandwaves, (Hulscher et al., 1993). As noted by Van Veelen et al. (2018) in their study of sandbank evolution, the evolution of the seabed follows from the tidally averaged bed load sediment transport, which is directly related to the tidal residual currents. Indeed, studies of the historical evolution of the Gt. Yarmouth sandbanks (Horrillo-Caraballo and Reeve, 2008) and sandbanks in the Pentland Firth (Chatzirodou et al., 2017) have confirmed the importance of this feedback mechanism in coastal morphodynamics. From this perspective, the analysis and interpretation of tidal residuals with respect to a single snapshot of bathymetry, especially one that is composed of mobile sediments, is best seen as a diagnostic tool rather than a reliable prognostic technique. Doodson et al. (1954) developed one of the earliest computational models of tides in the Irish Sea simulating the propagation of the M 2 tide and validated their results against the observations of Doodson and Corkan (1932). Subsequently, Robinson (1979) produced cotidal maps for the main constituents (M 2 , S 2 , N 2 , O 1 , K 1 ) and some shallow water constituents using current meter measurements of the area of the Irish Sea. One of the first computational models of the North-west European continental shelf was presented by Flather (1976). Modelling capabilities improved rapidly in the following decades as computer speeds and computational modelling techniques evolved. Davies (1986) and Davies and Jones (1992) considered three-dimensional effects in their model of main constituent tides in the Irish and Celtic Sea, which was based on a vertical eddy viscosity parameterisation technique. Results were improved by Aldridge and Davies (1993) and Davies and Aldridge (1993) with higher resolution grids and by Gekeler (1995) who incorporated data assimilation into a 3-D finite difference model of tides in the Irish Sea. While Jones and Davies (2007) investigated the contributions of the five main tidal constituents plus the overtides M 4 and M 6 using a finite element model, concluding that it was necessary to include shallow water harmonics to achieve good accuracy in the tidal elevations and currents. The asymmetry of the tidal variation can have significant implications on the efficiency of tidal stream energy extraction, as argued by Neill et al. (2014), as the power output is related to the velocity cubed, which means that small changes in asymmetry in the velocity will have a large effect on the power output. Additionally, construction of tidal energy infrastructure such as tidal barrages has the potential to alter the larger scale tidal propagation characteristics, (Zhou et al., 2014). The preponderance of studies has concentrated either on the Liverpool Bay area or on the Severn Estuary or in small areas suitable for marine energy device deployments. Areas such as Cardigan Bay have received little attention; in general, the tidal dynamics around the Welsh coast has not been studied as in an integrated and coherent manner. In this study, a hydrodynamic model of the Irish and Celtic Sea has been configured to investigate the barotropic tidal dynamics and the effects of the non-linearity of the tides (compound tides and overtides) around the coast of Wales. The effects of stratification on tides have been noted in localised regions within this domain, (e.g. Pingree, 1980), but are not considered in this paper. The model employed is Delft3D which was configured to run for depth-averaged flow. The tidal flows were driven by 13 constituents along its open boundaries, and the seabed is represented by high resolution digitised bathymetry. The model was run for a period of one year to determine the spatial distribution of the amplitude and phase of the predominant harmonics, tidal residual currents and their vorticity. Daily variations in the position of amphidromes, as reported by Pugh (1981), are not resolved over this period. The inclusion of more tidal harmonics, updated bathymetric information and long simulation time represent a significant enhancement of modelling fidelity. The aims of this paper are twofold: first, to provide a detailed analysis of tides around the Welsh coast; and second, to provide an integrated source of information on the tidal hydrodynamics of the area. The study area is described in Section 2. In Section 3 a description of the model, its calibration and validation are presented. Section 4 contains the results and discussions of the study and the paper finishes with a short set of conclusions in Section 5.

Study area description
The study domain covers the Irish Sea and part of the Celtic Sea; having significant areas where the tidal range can be classified as megatidal and is suitable for deployment of tidal range energy devices. Fast flowing currents are of interest for tidal stream energy extraction. The Celtic and Irish Seas are extremely important areas in terms of fish and invertebrate biodiversity, supporting a large diversity of seabirds and marine mammals, along with several important European fisheries [ICES, 2018).
The physical oceanography of the Irish Sea is driven mainly by tidal currents, with tidal streams of 1 m/s in magnitude and water tidal elevations up to 9 m (Robinson, 1979). Bowden (1980) defined the Irish Sea as comprising an area extending from Carnsore Point (Ireland) and St. David's Head (Wales) in the south to the North Channel between Larne and Corsewall Point to the North. Its width is around 75-140 km and length is approximately 150 km between Ireland and Wales. In its northern area, its largest extent is 195 km from east to west and 150 km from south to north, with the Isle of Man around the centre of the area. The mean volume of the Irish Sea has been estimated as 2430 km 3 and its area as 47,000 km 2 (Howarth, 2005). The run-off fresh water received by the Irish Sea comes from a large catchment area, 43,000 km 2 , mostly arriving from the Eastern Irish Sea (Ribble, Mersey and Dee estuaries; Solway Firth and Morecambe Bay). Also receives a significant contribution of fresh water from the Severn Estuary via the Bristol Channel. According to Howarth (2005) shallow estuaries are found in the area, such as Solway Firth, Morecambe Bay and the Dee Estuary. In addition, extensive sandbanks can be found to the north and east of the Isle of Man (Bahama and King William Banks) and off the Irish coast south of Dublin (Kish, Codling, Arklow and Blackwater Banks). At its southern boundary, St. George's Channel links the Irish Sea with the Celtic Sea. Shallow areas, of depths less than 50 m, can be found to southeast of Cardigan Bay and the east part area of the Isle of Man (Ozer and Legrand, 2015). The Irish Sea is constrained by two narrow channels so waves are predominantly locally generated, with short periods and are often steep. Swell waves in the area are only present near the entrances and southern end of the St. George's Channel. But they also can propagate as far as the Llyn peninsula (North Wales) and the northern part of the Northern Channel (Howarth, 2005). Cooper (1967) defined the Celtic Sea as a shallow embayment of the eastern North Atlantic Ocean surrounded by the south coast of Ireland, southwest Wales (UK), southwest England (UK) and Brittany (France). According to Pingree (1980), its area can be defined as bordered at the north with the Irish Sea and from the western entrance to the English Channel by a line drawn from Ushant (France) to Land's End (England). The border with the Atlantic Ocean is defined by the 200 m contour water depth (Pingree, 1980). Our high-resolution model grid covers the area shown in Fig. 1.
Tidal ranges at mean spring tides in the area vary between 2 m and around 12 m. Tidal current stresses on the seabed cause the suspension of fine material in the water column occurs due to the movement of sediments and due to the bottom friction of coarser material. The tidal stresses also cause the water column to mix, which can be a significant control on the seasonal thermal stratification in this area (Uncles and Stephens, 2007).

Delft 3D
In this study, we have employed the hydrodynamic modelling suite Delft3D. Delft3D is an open source three-dimensional (3D) model under active development led by Deltares (https://oss.deltares.nl/we b/delft3d). It computes solutions to the 3D Navier-Stokes equations using finite-difference approximations with a choice of turbulence schemes and gridding. Here, the model is configured to solve the depthaveraged flow as our primary interest is in the barotropic tidal dynamics. The derivation of the equations used in Delft3D, together with the computational solution method is well documented in the paper by Lesser et al. (2004). The model was configured with nested grids to provide higher resolution for shallow water regions.

Model domain definition
The hydrodynamic model is based on a set of two structured orthogonal curvilinear spherical grids nested to provide increased resolution in areas where the bathymetry, and flow pattern, is particularly variable. The two domain areas are shown in Fig. 2, in which the Continental Shelf Model (CSM) covers an area which is bounded by latitudes 40 • N and 60 • N and by longitudes of 20 • W and 12 • 30 ′ E and the Irish and Celtic Seas Model (ICSM) covers the area bounded by latitudes 55 • N (Glenarm, NI) and 49 • 30 ′ N and by longitudes of 5 • 12 ′ W (Lizard Point, UK) and 10 • W. The grid of the CSM model contains 590 × 506 cells and the grid spacing is approximately 3.5 km (0.03 • ) while the grid of ICSM contains 416 × 304 cells and the grid spacing is less than 2 km (0.0167 • ). The boundaries of the CSM were chosen to extend beyond the continental shelf, a factor known to be important for modelling the tides near the coast . For the ICSM, the grid covers the Welsh coastal waters including the Bristol Channel and estuaries.
The bathymetries for the CSM and the ICSM were taken from ETOPO (Amante and Eakins, 2009), GEBCO 2014 30 arc-second bathymetry (Weatherall et al., 2015) and EMODNET (EMODNET, 2016) and integrated into the model grids converting all levels relative to Mean Sea Level (MSL). A free surface condition was applied to the upper boundary and the bottom boundary was an impermeable bed with a standard formulation of bed shear stress. Specifically, the bed shear stress for the 2D depth averaged flow is induced by a turbulent flow and it is taken to be given by a quadratic friction law as follows, (Lesser et al., 2004):

Model calibration
Calibration of the model, specifically to determine the optimum value of the Chézy coefficient, requires comparison of model output against observations. The model generates surface elevations and depthaveraged tidal currents. Observed currents and elevations represent the combined effects of wind, waves and tides. Currents are traditionally measured at one or more discrete depths. To convert these into depth averaged currents against which to compare the results of a depthaveraged model would involve additional assumptions. Here, we focus on surface elevations to avoid such problems and there is no direct test of the instantaneous tidal currents produced by the model.
The computed tidal residual currents are compared against independent observations where the length of record allows surge and storm effects to be smoothed. Comparison of model output against observed surface elevations still has some uncertainties as the model excludes the effects of wind, waves and gradients in sea surface pressure. To mitigate these effects we compare instantaneous surface elevations from the model against observations over a two month period. In addition we compare the amplitude and phase of the dominant tidal harmonics computed from the model output against independent sources.
The calibration of the nested modelling system proceeded in four stages. First, the CSM (coarse grid) model was tested for sensitivity to grid resolution. Computations were performed at three grid resolutions, (5 km, 3.5 km and 1 km), in order to choose the best grid spacing. Computed tidal elevations over the two month period 1st January to March 1, 2003 were compared against measured elevations at all the observation locations shown in Fig. 3. The 3.5 km grid was the best option for these studies giving good results in reasonable computing time. Different time steps (0.25, 0.5, 1.0 and 2.0min) were tested to find an acceptable balance between stability and efficiency. A time step of 2 min was selected. In the second step a set of tests using the 3.5 km grid and 2 min time step were performed for a range of values of the Chézy coefficient. Modelled elevations were again checked against the observations to select the value of Chézy coefficient that gave the least error. The range of values of the Chézy coefficient used for the CSM testing was 40, 55, 60, 65, 75, 90 m ½ /s. Step three involved driving the ICSM with boundary values provided by the calibrated CSM, for grid resolutions of 3.5 km and 1 km and time steps from 0.25 min to 2 min. The optimum values being a resolution of 1 km and a time step of 2 min. The fourth step was to run the ICSM with resolution 1 km and time step 2 min for values of the Chézy coefficient from 60 to 90 m ½ /s. The calibration results for both models are presented together. Time series of water levels and tidal constituents at the observation points were taken from the tide station toolbox from Delft dashboard (Nederhoff et al., 2016). Measured water levels were recovered from the International Hydrographic Organization, (IHO). Three error metrics were used for the calibration: Root Mean Square Error (RMSE); Correlation coefficient; and percentage RMSE. All quantities were calculated over the results from the two month calibration model runs. Tabulated summaries of calibration statistics for the CSM may be found in Appendix A, Table A1, for the observation points (blue squares in Fig. 3). Tabulated summaries of calibration statistics for the ICSM may be found in Appendix A, Table A2.
Scatter plots for the CSM and ICSM, showing measured elevations versus modelled ones, are presented in Fig. 4 for six stations across the area. The locations are coloured blue in Fig. 3. In each plot, each point represents an hourly value; its x-ordinate being the observed elevation and the y-ordinate being the modelled elevation. The points are colourcoded according to the value of the Chézy coefficient used in the computation. In Fig. 4 the three locations on the left hand side of the figure show are fairly even distribution of points around the perfect fit, (y = x line). The three locations on the right hand side show a different behaviour. In particular, the spread of points for Hinkley Point is noticeably large, and there is a systematic deviation away from the y = x line for both Lundy Island and Fishguard. All three of these sites are in the south of the domain and exposed to south-westerly storms, which would contribute a considerable surge components to the measured elevations. Overall, the spread of points is small, indicating a good level of agreement between model and observations.
The values of the Chézy coefficient that provided the best calibrations when considered against all stations shown in Fig. 3 were C = 75 m ½ /s for the CSM and C = 60 m ½ /s for the ICSM. A slight change in coefficient between nested grids is not uncommon and reflects the scaledependent nature of the parameterisation of frictional processes. Overall, the ICSM grid gave improved results in comparison with those from the CSM grid.

Model validation
The process of validation of a model requires testing the calibrated model against a set of observations not used in the calibration procedure. This serves as an independent test of the model, providing additional assurance of its performance. Validation was undertaken over a period of one year, between the February 1, 2015 and the February 1, 2016. This is an extended test but was chosen in order to capture a large range of tidal harmonics. Forty-nine stations were considered in the validation. Modelled elevations at each of the stations for the validation period were subjected to harmonic analysis. Table 1 shows the RMSE and correlation coefficient between modelled elevations and the IHO data for the CSM and ICSM calculated over the longer validation period of 1 year at the same locations as used for the calibration. When comparing the errors against those in the calibration, (Table A2), it is evident that a very similar level of performance has been achieved against the validation data set.
As a more stringent test of the model than calculating errors between predicted and observed tidal elevations we have chosen to compare the results of an harmonic analysis of the elevations. Table 2 summarises observed and computed amplitudes and phases and their differences for each of the constituents at a selection of stations (Fig. 3). It is noted that some of the largest differences between model and observed phase occur at stations located near amphidromic points. This is not unexpected  Table 2 were calculated using the harmonic analysis tool within Delft3d. Table 2 demonstrates that the model reproduced amplitudes and phases along the shoreline to a good level of agreement with IHO data. By inference from continuity considerations it also provides a good estimate of the amplitudes and phases throughout the model domain. Results for the full set of 49 sites are provided in Appendix B. The harmonic amplitudes agree to within 16 cm or better and harmonic phases agree on average to 22 • or better across the observation sites.

Results and discussions
In this section a detailed description and analysis of tidal properties, including co-tidal charts, tidal asymmetry and tidal current ellipses around the Welsh coastline are presented. All results shown have been computed from the output of the validated ICSM.

Tidal properties
Based on the classification on tidal range proposed by Davies (1964), and subsequently extended by Levoy et al. (2000), much of the coastline along the Bristol Channel and Morecambe Bay can be classified as mega-tidal (tidal ranges > 8 m), the remainder being macro-tidal (4 m < tidal range < 8 m). In contrast, the tides along the coast of Ireland are meso-or micro-tidal (2 m < tidal range < 4 m and tidal range < 2 m respectively). Our model results confirm that the tidal range along the Welsh coast varies between mega-and macro-tidal. The tidal form factor, F, (Pugh and Woodworth, 2014), has values less than 0.25 throughout the study area, indicating that the tides in the Irish and Celtic Sea areas are strongly semi-diurnal. Fig. 5 shows the tidal range determined from the CSM. The spatial variation in the amplification of the tidal oscillation around the Celtic/Irish Sea is evident, with macro-and mega-tidal ranges evident around most of the Welsh coast.

Co-tidal charts
Co-tidal charts show the spatial variation of the amplitude and phase of an individual tidal harmonic, illustrating the relative contribution of that harmonic to the total tidal flow. Co-tidal charts, calculated from the ICSM tidal elevation output, for the main tidal harmonics O 1 , K 1 , N 2 , M 2 and S 2 are shown in Fig. 6. These are in good qualitative agreement with earlier results derived from point observations by Mungall and Matthews (1978), Robinson (1979) and Howarth (1990).
The charts in Fig. 8 also compare well in general terms with those obtained with coarser grid circulation models, such as Davies and Jones (1992), while providing additional detail.
Furthermore, the position of the amphidromic point of the M 2 harmonic near the Irish coast agrees well with the position suggested by the analysis of tide gauge observations presented by Robinson (1979) and Howarth (1990). The amplitudes in Fig. 6 clearly demonstrate that semi-diurnal tides are the significant constituents in this study area, and that the M 2 harmonic is the predominant harmonic.
One important feature of the two major semi-diurnal tides (M 2 and S 2 , Fig. 8) is that their amplitudes increase remarkably along the Welsh and English coasts but are damped along the Irish coast when they propagate into the Irish Sea. This can in part be explained by the Coriolis effect that acts to divert the tidal wave propagation path towards the

Table 2
Amplitude, A (in metres), and phase, ph (in degrees), from a selection of the IHO tidal stations included in Delft Dashboard (Nederhoff et al., 2016), model results and their differences.    right, but in the main by the resonance properties of this sea region. Specifically, an approximate calculation tells us that as the distance from Fishguard (St. George's Channel) to Stranraer (southern end of North Channel) is roughly 300 km and the mean depth of the sea between the two is close to 100 m the propagation speed of a long wave will be approximately 30 m/s. So if the tidal period is 12 h, a quarter wavelength region will have a length of about 300 km. The dimensions of the Irish Sea are almost perfect for creating a resonant response. This may explain in part the relative insensitivity of our results to small changes in Chézy coefficient, where other authors have found resonance phenomena to be sensitive to the choice of this parameter, (e.g. Gao and Adcock, 2017). In reality, the Irish Sea is not rectangular but of irregular shape, its bathymetry is not constant but varies considerably, and the progression of the tide wave is affected by Coriolis accelerations. In the North Sea the Coriolis effect causes the tidal wave to propagate in an anticlockwise sense around the shoreline, with largest amplitudes at the coast and an amphidrome towards the middle of the North Sea, (Howarth 1990). In contrast, the Irish Sea is not sufficiently broad to allow the tide wave to propagate around its edges without interference. Taylor (1919) found that the tidal movement within the Irish/Celtic Sea was a co-oscillating response of the shelf sea to the tides generated in the Atlantic Ocean. Taking the M 2 constituent as an example, its amplitude is about 3.6 m in the middle of the Severn Estuary (England and Wales) and decreases to less than 0.5 m in the Irish Coast (Rosslare area) and increase again towards the Liverpool Bay area to 3.0-3.3 m. Two amphidromic points (M 2 component) can be found in this area: one in the North Channel (not shown), and one "degenerate" amphidromic point (where cotidal lines join on a point inland), in St. George's Channel. The other main semi-diurnal harmonics, N 2 and S 2 , show a pattern similar to that of M 2 but with diminished magnitude. The diurnal harmonics, O 1 and K 1 , show a different pattern from the semi-diurnal harmonics as might be anticipated from their larger wavelength and period. The co-tidal charts (Fig. 8a,b,c) show a rapid increase in amplitude and phase from the Celtic Sea into the Irish Sea, with a maximum amplitude in the Severn Estuary and Liverpool Bay. According to Huntley (1980), in the Irish Sea the tide moves with a "rocking" motion where the occurrence of the high waters is alternates at the open and closed ends of the sea with a small tidal range in the central region. This behaviour is evident in the approximate 180⁰ phase difference in both M 2 and S 2 constituents between the northern and southern parts of the domain. That is, at the time of high water in the northern end of the Irish Sea, it is low water in the southern end. It suggests too, the existence of strong tidal streams in the central part of this region that is characteristic of standing-wave motion. Fig. 7 shows the co-tidal charts of the main shallow water harmonics (MS 4 , MN 4 , M 6 , M 4 , 2SM 6 , 2MS 6 , 2MN 6 ) determined from the ICSM, giving an integrated picture of these harmonics for the entire Welsh coastal waters. Our results for M 4 and M 6 show good general agreement with those of Davies (1986) who used a relatively coarse resolution three dimensional finite difference model of the Northwest European continental shelf to simulate the M4 harmonic, and also with the results of Jones and Davies (1996) who presented results for M 4 and M 6 . The higher grid resolution of our model reveals additional fine structure. It is also clear from Fig. 7 that in estuarine regions, such as Morecombe Bay, Solway Firth in Liverpool Bay and in the Severn Estuary, there are rapid changes in the amplitude of shallow water harmonics over small distances.
Of the shallow water harmonics, M 4 is the dominant one, followed by MS 4 , MN 4 , 2MS 6 , M 6 and 2SM 6 (Fig. 7). According to Andersen (1999) the M 4 tidal harmonic is the largest shallow water constituent in the northwest European Shelf region. This is confirmed at a regional level by our results that show M 4 has a mean amplitude of approximately 4.3 cm over the domain with values > 15 times larger at some locations such as the Severn Estuary, the Mersey Estuary and Morecambe Bay. However, there are localised exceptions to this general picture where, for example, MS 4 is of comparable or greater magnitude. The three harmonics M 4 , MS 4 and MN 4 all exhibit an amphidromic point to the northeast of the Isle of Man.
The harmonic MS 4 has a similar pattern to M 4 but with smaller amplitudes. MS 4 has a mean amplitude of approximately 3.7 cm over the domain with maximum values > 13 times larger at several locations in the Severn Estuary and an amphidromic point in Cardigan Bay. The harmonic MN 4 shows a similar pattern to those of M 4 and MS 4 . The MN 4 harmonic has average amplitude of 1.5 cm with amplitudes beyond 20 cm at the Dee Estuary and Severn Estuary.
The last of the more dominant harmonics considered here is 2MS 6 . This harmonic has a mean amplitude of 1.6 cm, with amplitudes exceeding 15 cm in the Severn Estuary and the Dee Estuary. The most significant shallow water harmonics in this region are M 4 , MS 4 , MN 4 (quarter-diurnal constituents) and M 6 , 2SM 6 , 2MS 6 , 2MN 6 (sixth diurnal). Shallow water harmonics arising from interactions with diurnal harmonics K 1 and O 1 tend to be weak. Friedrichs and Aubrey (1988) describe two measures of tidal asymmetry. The first is the total distortion factor, which is defined as the ratio of the amplitudes of M 4 and M 2 : A M4 /A M2 . If A M4 /A M2 > 0.01, significant distortion of the tidal wave is expected. The second is the tidal dominance factor which is defined in terms of the phases of the harmonics M 2 and M 4 : |2ϕ M2 − ϕ M4 |. If |2ϕ M2 − ϕ M4 | is between 0 • and 180 • , the tide is flood dominant and if it is between 180 • and 360 • , the tide is ebb dominant. The relevance of these to tidal energy generation and net transport of sediments may be viewed as follows. Tidal energy exploitation is concentrated on shallow continental shelves, exactly where shallow water harmonics may be largest. The presence of shallow water harmonics will produce asymmetries in the tidal oscillation and consequently power extraction, reducing the overall energy yield, (Neill et al., 2012). Similarly, asymmetries in the flood and ebb flows will lead to inequalities in the durations for which near bed currents exceed the threshold velocity and hence differences between the quantity, direction and distance of sediment transport on the ebb and flood tides, leading to a net drift, Prandle (2009) .  Fig. 8 shows the parameters for tidal distortion and asymmetry in the Irish and Celtic Seas. Fig. 8a shows that the areas where the tide is less distorted, (A M4 /A M2 < 0.01), are mostly outside the estuaries. The greatest tidal distortions are found in the estuaries (e.g., Severn, Dee, Mersey, Ribble and Alt, Solway Firth, Milford Haven, etc.). Also highly distorted tides are present along a section of the open Irish coast (from  Fig. 3). The transitions correspond to the nodal lines of a standing wave oscillation described in Section 4.2, which explains the large tidal range experienced along the Liverpool Bay coastal area.

Tidal asymmetry
From a dynamical perspective, a large distortion factor indicates frictional energy dissipation arising in shallow waters and transfer of energy from M 2 to M 4 , through nonlinear processes, (Aubrey and Speer, 1985;Friedrichs and Aubrey, 1988). The distortion of the tidal oscillation in estuaries may be understood from this perspective. However, off

Tidal ellipses
The contribution of each tidal harmonic to the depth-averaged tidal flow may be found using harmonic analysis, applied to each component of the horizontal velocity. The constants from each component of velocity may be combined to construct an ellipse that describes the tidal motion due to an individual tidal harmonic, (Howarth 1990) . Tidal ellipses encapsulate information about the maximum and minimum tidal current speeds and directions, the sense of rotation of the tidal current, and the flow in relation to the state of the tide. A long thin ellipse indicates almost rectilinear tidal flow, with a rapid switch between flood and ebb flows. In contrast, broad, almost circular ellipses indicate the tidal current speed is almost constant as the direction changes. The sense of rotation is affected by the geometry of the seabed, the shape of the coastline and the Coriolis effect. The latter will tend to create clockwise rotation (in the Northern hemisphere).
Time histories of the tidal currents computed in the ICSM over the one-year validation period were analysed to extract the harmonic constants. The tidal ellipse parameters for the M 2 tidal constituent at each grid point were calculated from the corresponding time histories. The results are shown in Figs. 9 and 10 (a-e) with red and blue denoting clockwise and anti-clockwise rotation respectively. Fig. 9 shows the general distribution of the tidal ellipses every ten grid points for the ICSM area, while Fig. 10 shows results for enlarged subregions.
Tidal ellipses are aligned along channel axes in: St George's Channel; the Bristol Channel; and the Northern Channel, indicating that rotation is relatively unimportant due to the geometrical constraints. The ellipses are very elongated, typically with their minor axis being less than 10% of their major axis. Broader tidal ellipses occur at the intersection of channels, due to transverse reflection effects, and in Morecambe and Cardigan Bays where the localised broadening of channels creates a more relaxed tidal flow. The general trend of the tidal ellipses for the M 2 constituent is similar to the tidal current circulation patterns presented in the previous section, and also reflects the change in tidal phase or 'rocking' pattern associated with the standing wave nodal lines discussed in Section 4.2.
Tidal ellipses close to the shore and to the north of Anglesey, Llyn Peninsula and Pembrokeshire have their major axis aligned with the coast and are highly elongated. Immediately to the south the tidal ellipses become more rounded, where there is no slack water. Both Anglesey and Pembrokeshire lie close to regions which switch from being ebb dominated to the north to flood dominated to the south, while the Llyn Peninsula is something of an exception as it experiences flood dominant tides to north and south. It is possible to find major to minor ellipse axis ratios of around 0.4 near Ramsey Island (Pembrokeshire, Figs. 10c) and 0.2 -0.3 near Anglesey and Llyn Peninsula ( Fig. 10a and  b). Fig. 10a shows that to the north of Anglesey, the ellipses indicate clockwise rotation of the tide; to the northwest there is a combination of clockwise and anticlockwise rotation and towards the west of Anglesey the tidal ellipses are more rectilinear and elongated and have anticlockwise rotation. Near to the Llyn Peninsula (Fig. 10b), the tidal ellipses show anticlockwise tidal rotation and are elongated; towards the west of the peninsula and to the south, the tidal ellipses show anticlockwise tidal rotation but have a tendency to be less elongated and more circular. Inside Cardigan Bay (Fig. 10c), most of the tidal ellipses show anticlockwise tidal rotation and are more circular than in the other  areas. Moving westwards out of the Bay, the tidal ellipses become more elongated than inside the bay, indicating transition to more rectilinear flow.

Tidal residuals 4.5.1. Residual currents
Residual tidal currents were calculated as grid point averages of instantaneous currents at each time step over the year-long validation run of the ICSM. The residual currents show a complex flow pattern, with several gyres in the area of the Eastern Irish Sea Basin, Cardigan Bay and the Welsh coast in the Severn Estuary. Plots of residual currents for selected areas are shown in Fig. 12 in the following section.
The residual currents are relatively weak, (<0.1 m/s), and around 0.01 m/s in most places. The general sense of the flow is from the Atlantic in the direction from south to north near the coasts with a more complicated pattern around St George's Channel. Part of the main flow diverts towards the Welsh coast moving northwards approaching the Isle of Man. One part enters the eastern Irish Sea (North of Anglesey) and gyres anti-clockwise near the Isle of Man rejoining the main flow northward to North Channel. In the middle of the Eastern Irish Sea an anti-clockwise gyre is apparent, which was also noted by Ramster and Hill (1969) in their study of the tidal circulation of the Northern Irish Sea using Woodhead seabed drifter and current meter measurements. Anti-clockwise gyres are also evident in Liverpool Bay and the outer Bristol Channel near Ilfracombe.
The magnitudes of the computed tidal residual currents are in good qualitative agreement with the study of Ramster and Hill (1969) for the Northern Irish Sea, and are in agreement with the patterns obtained by other studies such as the observational campaign of Robinson (1979) and the national scale studies of Brown et al. (2015) and Williams et al. (2019). The southward residual flow to the north of Anglesey is at variance to the results found by Prandle and Ryder (1989) who compared surface currents derived from radar measurements with depth-averaged currents from a computational model. They found reasonable agreement between the two for the M 2 harmonic but less so for the M 4 harmonic. The radar measurements indicated a northward residual current leading to unrealistic estimates of flow. The residual currents computed from our depth-averaged model are smaller and reflect the seabed geometry as they are unable to distinguish flow variations with depth.

Vorticity
The strength of the circulatory flow in residual current patterns may be measured conveniently by its vorticity. For horizontal flows, such as the residual currents of the depth-averaged flow, the vertical component of the vorticity, ξ, is the only non-zero component and may be written as: where the depth integrated residual current is (U, V) and x and y are local Cartesian coordinates running in west to east and south to north directions respectively. It follows from this definition that positive and negative vorticity correspond to anticlockwise (cyclonic) and clockwise (anticyclonic) circulations respectively. Prandle and Ryder (1989) presented measurements of coastal vorticity together with linkages between depth-averaged equations and vorticity. They concluded that the magnitude of the vorticity was of the same order as the accuracy of their calculations. Subsequent advances in modelling technology and computing power have led to the improvement of the accuracy of results. Calculation of the vorticity of tidal residuals has been presented, for example, by Neill (2009) and Yang and Wang (2013) to investigate sediment transport and water exchange in coastal waters. As noted in the Introduction, the analysis of residual currents and their vorticity is best viewed as a diagnostic tool for interpreting potential sediment transport trends, pollutant and water movements. The vorticity at each point of the grid was calculated using a centred finite difference scheme coded in MATLAB®, and included the additional metric terms associated with the spherical coordinate system used in Delft3D. Fig. 11 shows colour-flooded contour plot of the vorticity of the residual currents over the whole domain and Fig. 12 shows enlarged plots for Anglesey, Llyn Peninsula, Cardigan Bay and Pembrokeshire with the residual currents superimposed. Vorticity extrema in the residual flow are found in areas where sandbanks, headlands and abrupt changes to bathymetry are located. Residual flows in the Severn Estuary and estuaries around Liverpool Bay have large vorticity. In the Severn the pattern of predominantly positive vorticity along the English coast and negative vorticity along the South Wales coast is indicative of upstream flow along the edges of the estuary and downstream flow along the centre of the channel. There are also areas where the vorticity of the residual flow is close to zero, such as in Cardigan Bay away from the coastline and off the Irish coast between Rosslare and Donaghadee. Linear patches of positive/negative vorticity emanating from the south coast of Ireland into the Irish Sea coincide with sandbanks with similar dimensions. Typical values of the magnitude of vorticity are of the order of 10 − 5 to 10 − 6 s − 1 , which corresponds to the value found in other studies around the North Sea and the continental shelf, such as Nihoul and Ronday (1975);Zimmerman (1978), Zimmerman (1981) and Horrillo-Caraballo and Reeve (2008). Fig. 12a shows a large area of positive vorticity towards the western part Anglesey which corresponds to a major gyre in the residual current. This is attributable in large part to the amplification in the northwesterly residual flow while approaching the Anglesey coastline, rather than the formation of a strong circulatory flow. Shoreward of this gyre (northwest part of the Isle) is a region of negative vorticity. There are also many smaller residual eddies. On the northeast coastline of Anglesey, the vorticity is positive and to the northwest tends to be negative. An elliptical anticlockwise residual gyre is located on the west side of Anglesey.
Areas of positive and negative vorticity are apparent in Fig. 12b southwest of the tip of the Llyn Peninsula, west of Bardsey Island. A gyre with positive vorticity is clearly visible while the area of negative vorticity to its northwest also corresponds to a gyre which is slightly weaker. The formation of such a 'dipole' is readily understood from the arguments proposed by Pingree (1978), Zimmerman (1978) and Robinson (1983) who argued that flood and ebb flows would set up transitory eddies on the lee side of a promontory which, when averaged over one or more tidal cycles would yield a dipole pattern. Fig. 11. Vorticity field of the residual currents in the Celtic and Irish Seas.   12c shows the vorticity in Cardigan Bay. A complex structure is present in this area, mostly due to its shallow but highly variable bathymetry. A succession of positive and negative gyres is present in the central and southern part of the bay. Positive vorticity regions coincide with shallow water areas (Sarn Badrig patches, Cynfelyn patches, which are shallow subtidal reefs near the coast of Borth and Aberystwyth - Fig. 11) while negative regions coincide with deeper water areas or deep channels around higher relief seabed features. The largest gyre located in the central part of the bay offshore of Aberystwyth is cyclonic and another cyclonic gyre is apparent in front of Borth, north of Aberystwyth.
In Pembrokeshire (Fig. 12d) there is an area of strong positive vorticity west of Ramsey Island. South of St David's Head a region of negative vorticity is present which continues to the south to St Brides Bay until encountering the anticlockwise gyre towards the west. This is an area where strong currents are present and is a potential site for tidal stream device deployment. Fig. 12e, shows strong positive vorticity gyres at the south east of Swansea Bay and south east of Carmarthen Bay which coincide with shallow water areas (Scarweather sandbank in Swansea Bay and Nash sandbanks in Carmarthen Bay) and negative vorticity on each side of these sandbanks that coincide with deeper waters.
Coastal  Robinson (1983) in his study of the Irish Sea and in the study of the Wadden Sea by Ridderinkhof (1989), who attributed the small scale variations in the residual current field to the transfer of vorticity to the time mean or residual flow. This transfer can be most effective in zones where there is a transition from straight to curved depth contours, creating a gradient in the production of vorticity.
The pattern of tidal residuals currents and vorticity has broader potential implications. For example, there are gyres evident in the coastal area between Aberystwyth and Borth in Cardigan Bay. This area is the focal point of converging and diverging currents. An anticlockwise flow pattern occurs in front of the Aberystwyth coast, whereas a clockwise flow pattern appears in the south of the Aberystwyth coast. This area coincides with persistent erosion (Wood, 1971). The pattern of residuals does not prove a link but is suggestive and would indicate beach material disturbed during storms or washed out from the River Dyfi (North of Aberystwyth) on peak ebb tides might be taken offshore by residual currents and lost from the coastal cell, thereby contributing to erosion of the beach.
There is a wide range of tidal residual current magnitude with the strongest found in the Severn Estuary, sandbank areas in front of the Irish Coast (Arklow, Blackwater, Glassgorman Banks), north-west of Anglesey, Llyn Peninsula, in the North Channel and north and south of the Isle of Man, sandbanks in the Bristol Channel (Nash, Helwick, Scarweather Banks). Strong residual currents are found particularly near headlands, and near sandbanks. Areas of weak residual currents are found in the southwest of the Isle of Man, in the middle of the Irish Sea between the Irish coast and Cardigan Bay in Wales, also between the Isle of Man and the Cumbrian coast.
The sandbanks along the Irish east coast are areas identified for the development of offshore windfarms, (Arklow Banks wind farm) and the proposed Kish and Bray Banks wind farm. Knowledge of the residual flows around these areas is important for understanding likely sediment movements over the operational lifetime of the farms that might affect their stability and exposure to waves and currents. The banks offer coastal protection and they exercise a strong control on the tidal flow pathways near the coast. The residual current on the Arklow sandbank (north of Rosslare), reveals a clockwise circulation along the bank with a residual flow northward on the left side of the bank and a southward direction of the residual flow on the right side of the bank. This residual flow pattern circulation tends to maintain the sediment within the gyre, following the mechanism proposed by Huthnance (1973). Intensive 3-D modelling of residual currents and potential sediment movement may be justified in areas where tidal stream resources are planned, (e.g. Fairley et al., 2015;Chatzirodou et al., 2019).

Conclusions
A tidal model for the Irish Sea and the Celtic Sea with a resolution of ~2 km has been constructed, calibrated and validated against independent data. The model was nested from a coarser, shelf-scale model with a resolution of ~3.5 km driven by 13 tidal constituents: M 2 , S 2 , N 2 , K 2 , K 1 , O 1 , P 1 , Q 1 , MF, MM, M 4 , MS 4 and MN 4 . The validation of the model against independent tidal measurements demonstrated that the model is capable of correctly capturing the barotropic tidal dynamics of the Irish and Celtic Seas. Detailed analyses of tidal characteristics have been presented with discussions on how these may feed into broader considerations of managing the marine environment. The geographical variation in the tidal characteristics around the Welsh shoreline is large. The magnitude and distribution of this variation has implications for future tidal energy developments, coastal protection, flood defence and ecological studies. The findings presented here provide new knowledge required for such studies.
Our most important findings are: • The South-and North-Wales coastline is largely mega-tidal while the west coast is macro-tidal. The tidal regime is such that tides are semidiurnal everywhere and the M 2 tidal harmonic predominates; • A 180-degree phase difference in M 2 and S 2 tidal components exists between the northern Celtic Sea and the middle region of the Irish Sea confirming the picture of a standing wave type tidal motion; • Estuaries and semi-enclosed nearshore areas around the Welsh coastline experience significant tidal amplification. M 4 is the most significant shallow water harmonic around the Welsh estuarine coast and around the north-east Irish coast where the M 2 harmonic is diminished due to an amphidromic point; • The generation of M 4 and other overtides and compound harmonics around the Welsh coast means that the tides have a fine spatial structure and are asymmetric; • The south and north coastlines of Wales are largely flood-dominant while the west coast is ebb dominant; • The structure of tidal residual currents exhibits a rich pattern of gyres, the strongest of these being closely correlated to the locations of offshore sandbanks. The presence of the largest gyres may be understood from the generation and dissipation of vorticity of the depth-averaged flow.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. contribution that COASTal habitats make to human health and WEll-Being, with a focus on the alleviation of natural hazards (NE/N013573/ 1) project.