Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

DOI: 10.1007/s13131-011-0142-3

http://www.hyxb.org.cn

E-mail: hyxbe@263.net

Numerical investigation of wave propagation in the

Liverpool Bay, NW England

LI Ming1∗, RAYMOND Ip1, WOLF Judith2, CHEN Xueen3, BURROWS Richard1

1 School of Engineering, University of Liverpool, Liverpool L69 3GQ, UK

2 National Oceanography Centre, Joseph Proudman Building, Liverpool L3 5DA, UK

3 College of Physical and Environmental Oceanography, Ocean University of China, Qingdao266100, China

Received 23 January 2011; accepted 14 June 2011

©The Chinese Society of Oceanography and Springer-Verlag Berlin Heidelberg 2011

AbstractThe computer model for near shore wave propagation, SWAN, was used to study wave climates inLiverpool Bay, northwest England with various input parameters, including bottom friction factor,white capping, wind drag formulation and effects of tidal modulations. Results were comparedwith in-situ measurements and reveal the impacts from these inputs on the predictions of waveheight and propagation distributions. In particular, the model results were found very sensitive todifferent input formulations, and tend to underestimate the wave parameters under storm conditionsin comparison with the observations. It is therefore important to further validate the model againstdetailed field measurements, particularly under large storms that are often of the primary concern.

Key words: coastal engineering, flooding, wave model, near shore, tide

1 Introduction

The study of wave climate is important sincewaves play a pivotal role in sediment transport inthe nearshore zone, which is a major contributory fac-tor of coastal erosion. It also determines the possibleextent of coastal flooding during storms which maycause catastrophic consequences if coastal defences areovertopped. Waves in shallow water are strongly con-trolled by the water level as well as the wind forcing. Inparticular, the wave propagation within Liverpool Bayhas considerable impacts on the coastline of northwestEngland, with particular regard to the construction ofsea defences, coastal flooding and sediment transportalong the coast. Detailed knowledge of the prevailingwave climate is therefore regarded as an essential stepfor further assessment of shoreline behaviours. Thusfar, National Oceanography Centre (Liverpool) hasestablished several monitoring sites using the latestfield measurement technology, including wave buoys,acoustic doppler current profiler (ADCP) and high fre-quency (HF) radar through coastal observatory pro-gramme (Fig. 1). These data provide valuable basisto verify computer models and reveal important physi-

cal processes that can provide in-depth understandingof the wave climate within Liverpool Bay. By makinguse of these data and a nearshore wave model SWAN,the occurrence of typical and extreme wave conditionsin Liverpool Bay and the adjacent estuaries could beinvestigated. It is instrumental in assessing the risksin areas which may be susceptible to flooding due towaves in combination with high water levels. In addi-tion, through studying the phenomenon of tidal mod-ulation, more thorough understanding could be gainedon the actual physics and possible effects of tide andsurge on the nearshore wave parameters so that engi-neers could review the adequacy of design of existingcoastal structures.

When the SWAN is applied to the UK coastalsites (Wolf, 2003), it is worth noting that varying thebottom friction formulation can have a significant ef-fect on the modelling results, with the Madsen for-mulation (Madsen et al., 1988) better than the JON-SWAP formulation (Hasselmann et al., 1973). Withthe SWAN model, it is possible to transform the statis-tical extreme events for extreme value analysis. How-ever, the interpretation of transformed event can beproblematic. To study future climate scenarios, it is

Foundation item: The Public Science and Technology Research Funds Projects of Ocean under contract Nos 200905001 and201005019; this work is partially sponsored by Engineering and Physics Science Research Council (UK) through DTA trainingscheme.

∗Corresponding author, E-mail: mingli@liv.ac.uk

1

2 LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

Fig.1. Map of Irish Sea with the locations of wave buoys and estuaries and red box enclosing Liverpool Baymodel area.

considered more preferable to choose the most likelyand worst case scenarios. Moreover, the coastal waveheight is limited by the water depth so it is not aslikely to increase even if offshore waves increase.

In order to ascertain the reliability of the SWANmodel, simple parametric modelling for the waveheight, the wind speed and the peak period in theLiverpool Bay was undertaken and compared with theresults generated from the SWAN (Wolf, 2003). Theparametric model serves as a rule to guide the long-term morphological model whereas the SWAN modelis regarded as producing the real wave field. The re-sults show that even with the correction to the waveheights, there is still a large discrepancy between theparametric and SWAN models in the nearshore zonewith a water depth below 10 m. This is because thecomplexity of the model like dissipation processes, bot-tom friction and depth-limited breaking in this regioncan not be easily parameterized. That is why an inves-tigation using a detailed model in the SWAN is oftenrequired.

The SWAN wave model has also been used tostudy the influence of climate change at the coast(Hargreaves et al., 2002). It is found that there is nota simple relationship between the offshore wave heightclimate and the inshore climate in a particular region.Important complicating factors are bathymetry, tidalrange and incident wave angle. To improve accuracy,

it is necessary to have better observations of nearshorewaves and a more thorough understanding of the jointprobability distribution of water level and wave height.

2 Current wave climate in the Liverpool Bay

The wave climate is usually described by the prob-ability of occurrence of a range of wave parameters,especially height, period and direction, and also pa-rameters such as frequency bandwidth which gives anindication of the bimodality, i.e., contribution fromwind sea and swell (Wolf, 2008a). The wave cli-mate can be comprised of three parts: the long-termmean climate, the annual or seasonal cycle and thenon-seasonal variability on both intra-annual and in-terannual time-scales. In the UK waters, the waveclimate is strongly seasonal with mean wave heightspeaking around January, but with a high risk of bothhigh monthly mean wave heights and extreme waveheights throughout autumn and winter (October toMarch). There is also high interannual variability inthe monthly mean wave heights (Wolf, 2008a).

The Liverpool Bay mainly refers to the area in thenorthern Irish Sea including the Mersey, Dee and Rib-ble Estuaries. It is relatively shallow with a maximumdepth of less than 40 m. Wind data in the LiverpoolBay are recorded on Hilbre Island as part of theNOC Coastal Observatory. There are three ways of

LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13 3

collecting wave data, namely, the Directional Wa-verider in WaveNet network, POL Triaxys wave buoyin themouth of the Dee Estuary and a phased-arrayHF radar system at Llandulas and Formby Point(Wolf, 2008b).

The wave data gathered for this study was takenfrom the WaveNet buoy in the Liverpool Bay area(53◦32.14′N and 3◦21.36′W) as shown in Fig. 1. Themean water depth at this WaveNet site is about 22m (LIVBAY Coastal Observatory). The wave datawere recorded at 30 min intervals during which thesea-state intensity was assumed to be constant. Sixyears of data (from 2003 to 2008) were used for anal-ysis. In this analysis, it is assumed that the global orregional weather patterns are not changing in a waywhich affects wave generation. Pictorial summaries ofthe frequency distribution of wave direction and waveheight measurements are given in the form of waverose in Fig. 2.

Fig.2. Wave rose for WaveNet buoy from

2003 to 2008.

Unlike the WaveNet buoy, the Triaxys buoy is lo-cated in a relatively sheltered region near Hilbre Islandat the mouth of the Dee Estuary. The water depth isshallower and the location is closer to the shore. Thissite is less susceptible to large wind-generated waves asthe shallow water depth and the presence of east andwest Hoyle sand banks cause wave breaking and re-fraction, thus protecting the site from severe wave at-tack. Similar wave analysis was carried out for Triaxysbuoy data. Four years of wave data (between 2004 and2007), collected at hourly intervals, were used to de-rive the wave statistics. The wave rose plot in Fig. 3

indicates that the majority of waves propagate alongthe NW and WNW directions with a wave height ofless than 2 m.

Fig.3. Wave rose plot for Triaxys data from

2004 to 2007.

3 SWAN wave model

The SWAN wave model is based on the spec-tral action balance equation rather than the energybalance equation and in terms of relative radian fre-quency σ rather than absolute radian frequency ω

(Holthuijsen, 2007). All information about the seasurface is contained in the wave variance spectrum orenergy density E(σ, θ), distributing wave energy over(radian) frequencies σ (as observed in a frame of ref-erence moving with current velocity) and propagationdirections θ(the direction normal to the wave crest ofeach spectral component). Wave models determine theevolution of the action density N(x, y, t;σ, θ) in space(x, y) and time (t). The action density is defined asN = E/σ and is conserved during propagation in thepresence of ambient current. For small-scale computa-tions, the formulation in Cartesian coordinates is givenby

∂N

∂t+

∂cxN

∂x+

∂cyN

∂y+

∂cσN

∂σ+

∂cθN

∂θ=

St

σ. (1)

The first term on the left-hand side representsthe local rate of change of the action density in time.The second and third terms denote the propagation ofthe wave energy in the two-dimensional geographicalspace. The fourth term represents the effect of shiftingof the radian frequency due to variations in depth and

4 LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

mean currents. The fifth term represents the depth-induced and current-induced refraction. The right-hand side contains St, which is the source/sink termthat represents all physical processes which generate,dissipate, or redistribute wave energy.

For large-scale computations, including those onglobal scales, Eq. (1) can be formulated in terms ofspherical coordinates with longitude λ and latitude ϕ.

∂N

∂t+

∂cλN

∂λ+ cos−1ϕ

∂cϕcosϕN

∂φ+

∂cσN

∂σ+

∂cθN

∂θ=

St

σ. (2)

For modelling coastal waters in the SWAN, thefollowing wave generation and dissipation processescontribute to St:

St = Sin + Sn/3 + Sn/4 + Sds,w + Sds,b + Sds,br, (3)

where Sin is the wave growth by the wind; Sn/3 isthe nonlinear transfer of energy by triad wave-waveinteraction; Sn/4 is the nonlinear transfer of energy byquadruplet wave-wave interaction; Sds,w is the wavedecay due to whitecapping; Sds,b is the energy dissi-pation due to bottom friction; and Sds,br is the energydissipation due to depth-induced wave breaking. De-tails of each term can be found in Holthuijsen (2007).

With a view to obtaining a numerically stable so-lution for the wave parameters to avert excessive runtimes, the SWAN employs an implicit scheme of thefinite difference method. As mentioned previously, thegoverning equation in the SWAN is the spectral actionbalance equation [Eq. (1)]. For instance, a rectangulargrid is chosen with constant mesh sizes of Δx and Δy

in x and y directions respectively. The spectral spaceis divided into elementary bins with a constant direc-tional resolution Δθ and a constant relative frequencyresolution Δσ/σ. Time discretisation is carried outwith an implicit scheme as discussed in Holthuijsen(2007).

In the current study, different combinations oftide, surge, wind, water depth and bed friction will berun in the SWAN model. The SWAN model would beestablished for the Liverpool Bay area in the easternIrish Sea including the Dee, Mersey and Ribble Estu-aries. The location of area to be modelled is shown inFig. 1. The first phase of SWAN modelling consists ofsetting-up a fine-scale two-dimensional coastal modelwhich runs on structured grids with a grid resolutionof 185 m in both x and y directions. It is noted thatthe wave set-up, which is a major output parameter of

the model, cannot be computed accurately in spheri-cal coordinates (Delft University of Technology, 2006).Thus, all stationary cases to be proposed under thisstudy will be run on Cartesian grids. The origin ofthe model grid (bottom left hand corner) is located at290 481 m eastward and 365 998 m northward. Thereare 387 grids in the x direction and 378 grids in the y

direction. In the frequency domain, there are 33 fre-quencies (0.05–1.03 Hz) and 36 directions.

In addition to the steady state mode, the SWANallows users to include the time-varying variables suchas water level, current and wind field in a non-stationary model at specified time steps. The outputdata of water level and tidal current from an oceano-graphic model developed in NOC, POLCOMS, havebeen used to investigate the influence of tide and cur-rent on waves over time. As the POLCOMS data aredefined in spherical coordinates, such a coordinate sys-tem will be used in the non-stationary model runs.The bathymetric dataset was obtained from POL onthe (1/400)◦ by (1/600)◦ longitude-latitude grid whichcorresponded to 185 m. The origin of the grid (bottomleft-hand corner) is located at approximately 3.663 75◦

(longitude) and 53.179 166 66◦ (latitude). The totalnumber of grid points is 435 and 383 in longitude andlatitude directions respectively. During the trial runs,it was found that there was a major convergence prob-lem in running a model in the Liverpool Bay using thelatest version of SWAN 40.72. Hence, an old version40.51 has been adopted in this study and the resultsshow a similar accuracy as compared with the newversion of the model in other sites.

With a view to appreciating the wave conditionsalong the coastline of the Liverpool Bay under var-ious climate scenarios, a number of sampling pointswill be selected for outputting the model results. Theoutput from the wave model is instrumental in assist-ing engineers in assessing the likelihood of floodingat those major coastal towns/cities in the LiverpoolBay. Therefore, the sampling points should be lo-cated in flood-prone areas along the coastline. Accord-ing to the flood maps available on the Environmen-tal Agency’s website (Environmental Agency, 2009[Q/OL]), the areas most vulnerable to flooding in-clude Rhyl, Point of Ayr, Flint, Neston, North Wirral,Ellesmere Port, Formby Point, Southport, Lytham StAnnes and Blackpool. These areas together with theWavenet and Triaxys buoys will become the outputpoints for the model results. Their respective coordi-nates are shown in Table 1. Meanwhile, output is also

LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13 5

defined along a line from North Wirral to the offshoreboundary to ascertain the variation of wave parame-ters as it propagates from deep water to shallow water(Fig. 4).

Fig.4. The location of field sampling points

and output line.

The JONSWAP spectrum has been imposed as aboundary condition in this model. The spectrum iswritten as:

Sη(f) =αg2

(2π)4f5exp

(− 1.25

f4p

f4

)γq, (4)

where α = 0.076( gF

U210

)−0.22

in which U10 is the mean

wind speed at 10 m above sea level, F is the fetch

length; fp =3.5g

U10

( gF

U210

)−0.33

; γ is peak shape param-

eter. The maximum fetch in the Liverpool Bay is lessthan 250 km (Wolf, 2008b) and the water depth inthe model area usually does not exceed 45 m as shownin Fig. 5. Based on these conditions, the offshorewave height (Hs) and peak wave period (Tp) at theboundary could be determined using the parametricwave modelling equation derived by Hurdle and Stive(1989) for fetch-limited wave growth:

Hs = 0.25U2

a

gα

√tanh

(4.3 × 10−5F

α2

), (5)

Tp = 8.3Ua

gβ3

√tanh

(4.1 × 10−5F

β3

), (6)

where α = tanh(0.6d0.75

); β = tanh(0.76d0.375

). The

nondimensionalised depth and fetch are d =hg

U2a

and

F =Fg

U2a

.

Table 1. Coordinates of field sampling points

Sampling pointLongitude-Latitude grid

north Latitude(◦) west Longitude(◦)

Rhyl 53.323 040 32 –3.498 486 16

Point of Ayr 53.356 886 14 –3.317 107 70

Hilbre Island 53.356 886 14 –3.231 196 16

Flint 53.259 091 11 –3.132 375 93

Neston 53.281 218 04 –3.075 559 87

North Wirral 53.416 930 46 –3.125 986 53

Ellesmere Port 53.294 651 14 –2.859 084 99

Formby Point 53.556 807 71 –3.104 593 22

Southport 53.652 341 77 –3.020 287 21

Lytham St. Annes 53.740 410 66 –3.022 419 89

Blackpool 53.814 974 80 –3.057 967 56

Wavenet Buoy 53.540 844 19 –3.360 992 67

Fig.5. Water depth in the Liverpool Bay.

4 Model validation

Before the proposed wave scenarios are modelledin the SWAN, it is vitally important to ensure that themodel can predict wave propagation and transforma-tion in the study area with a reasonable accuracy. Indoing so, test cases will be conducted to evaluate andverify whether the model results would closely matchwith the field observations. A storm event on 18 Jan-uary 2007 was chosen for model validation. Wave data(significant wave height (Hs) and mean wave period(Tm2) at WaveNet and Triaxys buoys collected duringthe third week of January 2007 was plotted in Figs 6and 7.

As illustrated in Fig. 6, the wave heights recordedat the Triaxys buoy tend to be lower than that at theWaveNet buoy. This is attributable to the shallowerwater depth and presence of sand banks near Hilbre

6 LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

Island, inducing more wave breaking activities. Figure7 shows a similar trend as in Fig. 6 despite a numberof abnormal sharp peaks at the Triaxys buoy.

Comparison between the computed and observedwave heights and periods are listed in Tables 2 and 3for both WaveNet buoy and Hilbre Island sites at highwater and low water on the 18th January 2007. The

Fig.6. Comparison of wave height at Wavenet

and Triaxys buoys.

Fig.7. Comparison of mean wave period at

Wavenet and Triaxys buoys.

overall agreements in wave parameters in deeper wa-ter at the WaveNet buoy are generally good. However,there are some discrepancies between the modelled re-sults and the field data in shallower water at the Tri-axys buoy, which can be explained by the fact thatthe presence of current, change in tidal level, spatialchange in bed friction and morphodynamic feedback

Table 2. Comparison of predicted wave heights and periods at Wavenet buoy and Hilbre Island with observed values

using different formulations at high water (5.81 m above ordinance datum), wind speed at 22.1 m/s and 250◦ north on

the 18th January 2007

Liverpool Bay Wavenet buoy Hilbre Island Traixys buoy Formulations for physics options

Hs/m Tm2/s Tp/s Hs/m Tm2/s Tp/s wind input whitecapping bed friction

Observation 4.23 6.56 8.33 2.53 4.98 8.15 — — —

Test 15 4.36 6.31 8.7 2.64 4.69 8.7 Janssen Janssen JONSWAP

Test 16 4.43 6.33 8.7 2.7 4.61 8.7 Janssen Janssen Collins

Test 17 4.66 6.4 8.7 2.55 4.3 5.48 Janssen Janssen Madsen

Table 3. Comparison of predicted wave heights and periods at Wavenet buoy and Hilbre Island with observed value

using different formulations at low water (–2.40 m above ordinance datum), wind speed at 18.2 m/s and 260◦ north on

the 18th January 2007

Liverpool Bay Wavenet buoy Hilbre Island Traixys buoy Formulations for physics options

Hs/m Tm2/s Tp/s Hs/m Tm2/s Tp/s wind input whitecapping bed friction

Observation 3.21 5.71 8.33 0.8 3.17 3.28 — — —

Test 18 3.29 5.68 7.94 0.64 1.88 2.39 Janssen Janssen Jonswap

Test 19 3.39 5.76 7.94 0.63 1.85 2.48 Janssen Janssen Collins

Test 20 3.49 5.66 7.94 0.67 1.86 2.62 Janssen Janssen Madsen

plays an important role in the wave transformationprocess, especially in the shallow water region. How-ever, they have not been taken into account in thisvalidation exercise due to the constraints of the modeland the time availability. Although the real situationscannot be truly reflected and reproduced, the accuracyof the model is considered acceptable.

5 Model application

To ascertain and predict the likely consequences

under different environmental conditions, the follow-ing present-day wave climate and climate change sce-narios have been devised and presented in Table 4 be-low, in which the highest astronomical tide (HAT),the mean high water springs (MHWS), the mean highwater neaps (MHWN), the mean low water springs(MLWS) and the cumulative steepness Method (CSM)are described in van Vledder and Hurdle (2002). Totest the model sensitivity to input parameters, dif-ferent source terms will also be tested in the SWANmodel. The input values of tidal level, surge, wind

LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13 7

speed and directions in the model are derived from thewind, wave and water level analyses performed in pre-vious sections.

Cases 1, 2 and 3 are designed to examine the ef-fect of different wind speeds. Cases 1, 6 and 14 willstudy the effect of wind direction on wave generation.Cases 4 and 5 are strong winds on spring tides, athigh water and low water, for illustration of the effectof tidal levels. Cases 1 and 7 reveal the effect on wavedue to the presence of surge. Cases 8, 9 and 10 areintended to simulate the future climate change. A cli-mate change scenario would be defined by means ofan increase in sea level and storminess with the windspeed increased by 10% and occurrence of 1 in the 100a water level. After taking into account the recommen-dations of NOC and DEFRA (DEFRA, 2006), proba-ble scenarios of sea level rise of 0.5, 1.0 and 1.5 m overthe next 100 a are chosen. Cases 1, 11, 12 and 13 willbe compared to ascertain the combined effect of differ-ent surges and tidal levels on the wave height. Cases15 to 20 will help to examine the effect of different for-mulations of source terms on the wave characteristics.Through the modelling of these case scenarios, a set ofoutputs is obtained which includes the parameters ofthe significant wave height, the mean wave period, thepeak period, the water depth, a root-mean-square or-bital velocity at the sea bottom, a wave-induced setup,and a energy dissipation due to the wave breaking and

the bottom friction.In Figs 8–11, the significant wave height is found

to increase monotonically with the wind speed. Forthe wind-generated waves in the Liverpool Bay, thewind is the primary driving force for the wave prop-agation. It is reasonable to assume a constant windfield in each case as the stationary run only considersa snapshot of environmental conditions. According tothe Shore Protection Manual (Coastal Engineering Re-search Centre, 1984), the wave height is directly pro-portional to the wind speed squared whereas the waveperiod is directly proportional to the wind speed. Thehigher wind speed implies a greater wave height anda longer wave period. Since the wave setup is propor-tional to the wave height at breaking, the larger waveheight would give a larger wave setup.

Apart from the wind speed, there are various fac-tors controlling the wave height such as wind direction,water level and source terms. In Fig. 12, the windblows from three directions, which are 135◦ (SE), 293◦

(WNW) and 315◦ (NW) respectively. The SE wind oc-curs very frequently in the Liverpool Bay. However,the wind speed blowing from land tends to be lowerand the wave growth is severely limited by a very shortfetch. Similarly, the fetch for the wave from NW islimited by the Isle of Man, while the wind-generatedwave from WNW has the largest wave heightbecause the fetch in this direction is the longest in the

Table 4. Case studies of wave climate scenarios

Water Wind Wind Boundary Boundary

Case level speed direction wave height peak wave Remarks

/m /(m·s−1) /(◦) /m period/s

1 5.80 20.0 293 6.8 11.6 HAT plus 0.5 m surge plus wind speed 20 m/s (99%) (θ=293◦)

2 5.80 16.7 270 5.7 10.7 HAT plus 0.5 m surge plus wind speed 16.7 m/s (95%)

3 5.80 14.8 270 5.0 10.1 HAT plus 0.5 m surge plus wind speed 14.8 m/s (90%)

4 4.39 23.1 260 7.8 12.4 MHWS plus max 12hr mean wind speed

5 –3.83 23.1 260 7.8 12.4 MLWS plus max 12hr mean wind speed

6 2.20 8.6 135 1.0 3.7 MHWN plus median wind speed 8.6 m/s from SE

7 5.30 20.0 293 6.8 11.6 HAT without surge

8 6.40 34.6 293 10.5 14.5 1 in 100 a water level plus 0.5 m long-term sea level rise

9 6.90 34.6 293 10.5 14.5 1 in 100 a water level plus 1 m long-term sea level rise

10 7.40 34.6 293 10.5 14.5 1 in 100 a water level plus 1.5 m long-term sea level rise

11 4.30 31.0 293 9.7 13.9 Extreme storm with surge of 2.2 m at MHWN

12 6.30 31.0 293 9.7 13.9 HAT plus 1 m surge (1 in 10000 a water level)

13 3.10 15.0 293 5.1 10.2 Moderate surge of 1 m at MHWN

14 5.80 20.0 315 6.8 11.6 HAT plus 0.5 m surge plus wind speed 20 m/s (θ=315◦)

15 5.80 20.0 293 6.8 11.6 Different wind input (komen vs janssen)

16 5.80 20.0 293 6.8 11.6 Different bed friction (madsen vs JONSWAP)

17 5.80 20.0 293 6.8 11.6 Without depth-induced wave breaking

18 5.80 20.0 293 6.8 11.6 CSM as whitecapping formulation

19 5.80 20.0 293 6.8 11.6 Triad wave-wave interactions switched on

20 5.80 20.0 293 6.8 11.6 Increased bed friction JONSWAP coefficient is 0.085

8 LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

Liverpool Bay (about 218 km).Figures 13 and 14 illustrate the effect of tidal level

and presence of surge on the wave height. It is foundthat the wave height increases with the water level.The wave height at the MHWS is greater than that

at the MLWS. In the presence of a higher surge (say2.2 m), the wave height obtained is greater than thescenario with a lower surge (1 m). At a higher waterlevel, the water depth is greater and therefore allowsthose waves with larger wave heights to approach the

Fig.8. Distribution of wave height. a. Wind speed=20 m/s and b. wind speed=14.8 m/s.

Fig.9. Distribution of wave period. a. Wind speed=20 m/s and b. wind speed=14.8 m/s.

Fig.10. Distribution of wave setup. a. Wind speed=20 m/s and b. wind speed=14.8 m/s.

LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13 9

Fig.11. Effect of wind speed on wave height.

Fig.12. Effect of wind direction on wave height.

shore without breaking. It is noted that the waveheight predicted at the output points in the Ribble,Mersey and Dee Estuaries is considerably lower thanthat off the coast. This can be explained by morewave breaking activities in the estuaries and presenceof features such as ebb shoal, flood shoal and sand barsat the mouth of the estuaries which reduce the waterdepth.

To ascertain the effect of the sea level rise, threedifferent future wave climate scenarios have beenshown in Fig. 15. In these cases, the long-term sealevel rise over the next 100 a is predicted to be 0.5, 1.0and 1.5 m respectively. Owing to the climate change,the water level will reach 1 in the 100 a level more fre-quently. To make allowance for the increased stormi-ness, an extreme wind speed of 34.6 m/s is assumed,

10 LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

Fig.13. Effect of tidal level on wave height.

which is about 10% higher than the maximum ob-served wind speed from 2005 to 2008. The resultsshow that the wave height increases with the sea levelrise. It could reach up to 3 m in North Wirral andFormby Point, 4 m in Rhyl and more than 5 m inBlackpool. At the Wavenet buoy, the predicted waveheight is nearly 10 m, which is much greater than the200 a wave height (7.99 m). It is because the long-termsea level rise has not been considered in the calcula-tion of 200 a wave height previously. In Fig. 15, it canalso be observed that the shallow water regions nearthe estuaries such as Neston, Southport and LythamSt Annes are particularly sensitive to the long-termincrease at the sea level. The modelled wave heightsshow an increase of about 30% when the sea level riseincreases from 0.5 to 1.5 m. In addition to the above,

it is worthwhile studying the combined effect of dif-ferent tidal and surge levels on waves. The extremestorm surge in the Liverpool Bay can be as high as2.2 m (e.g., on 18 January 2007) with a return periodgreater than 50 a. The surge elevation in Liverpoolis often the highest on the west coast of England andWales. At high tide like the HAT, the normal surge isonly 0.5 m without a severe storm. It is a 1 in 10 000a event to have a storm surge of 1 m at the HAT, i.e.,6.3 m above the ordinance datum, in Liverpool. FromFig. 16, it can be seen that the largest wave height willbe predicted for a surge of 1 m at the HAT. It reaches3.5 m in Rhyl and 4.9 m in Blackpool. It is interestingto note that even such a rare event will not producea wave height as high as that caused by the sea levelrise of 0.5 m, let alone the case with a rise of 1.5 m.It reflects how serious the consequence can be witha drastic sea level rise in future. In Fig. 16, it canalso be noted that the extreme surge of 2.2 m doesnot produce the largest wave height. The reason isthat such an exceptionally large storm surge typicallyoccurs at low- to mid-tidal levels, due to interactionsbetween the tide and the surge. Therefore, the totalwater level, which controls the wave breaking process,is not particularly high.

The depth-induced breaking is a very importantphysical process to be modelled for the wind-generatedwave. Like the bed friction, the depth-induced break-ing plays a crucial part in dissipating wave energy in

Fig.14. Effect of surge on wave height.

LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13 11

Fig.15. Effect of long-term sea level rise on wave height.

Fig.16. Effect of long-term sea level rise on wave height.

the shallow water. To appreciate its significance, twomodel runs have been made with and without break-ing function being activated. As seen in Fig. 17,the modelled wave height in the absence of depth-induced breaking appears to be much larger than thatwith wave breaking. The difference is particularly sig-nificant at Formby Point, Southport and Lytham St

Annes where the water depth is very shallow. With-out wave breaking, the wave heights can grow tounrealistically high values. In some cases, they areeven greater than the water depth which is physi-cally impossible. Hence, this process has to be ac-tivated and taken into account in the coastal wavemodelling.

12 LI Ming et al. Acta Oceanol. Sin., 2011, Vol. 30, No. 5, P. 1-13

Fig.17. Effect of depth-induced breaking on wave height.

6 Conclusions

This study is an in-depth investigation on thewave climate in the Liverpool Bay using the SWANwave model. The primary research focus includes thepossible consequences of present-day wave climate andclimate change scenarios, tidal modulation, effect ofwind, surge, tidal current and various source terms onthe wave characteristics.

The existing wind data from Hilbre Island, wavedata from WaveNet and Triaxys buoys, and tidal datafrom Gladstone Dock have been analysed to work outappropriate input values for subsequent case studies.The SWAN fine-scale two-dimensional model has beenset up which runs on structured grids with a grid res-olution of 185 m×185 m. Model validation is carriedout and the optimum formulations of source terms aredetermined. The results show that the wave height,the mean wave period and the wave setup increasewith the wind speed. The wave height is affected bythe wind direction and the available fetch. Increasein the tidal level, the presence of surge and long-termsea level rise contribute to a higher water level. Theresulting wave height tends to be greater, posing aflooding risk to the coast. The wave height is sensitiveto the formulations of source terms. For the bed fric-tion, the JONSWAP formulation provides less energydissipation than Madsen. For the wind input, thereis no significant difference on the wave height. The

wave height increases when the bed friction is reducedand the depth-induced wave breaking is deactivated.In terms of whitecapping, the CSM formulation dis-sipates more energy than Janssen. At the low wa-ter spring tide, the largest setup in the Liverpool Bayranges from 0.2 to 0.35 m and is mainly concentratedalong the coast. This is because the shallow waterdepth induces a larger setup. The largest wave heightoccurs with a sea level rise of over 0.5 m in the next100 a. It is even higher than the wave height pro-duced by a storm surge of 1 m at the HAT which isa 1 in 10 000 a event.

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