Nesting a nearshore wave model (SWAN) into an Email: wlu@pol. … · 2014. 5. 13. · Nesting a...

Nesting a nearshore wave model (SWAN) into an

ocean wave model (WAM) with application to

the southern North Sea

Weimin Luo and Roger FlatherProudman Oceanographic Laboratory, Bidston Observatory, Birkenhead,

Email: wlu@pol. ac. uk

Abstract

This paper presents a numerical scheme to nest a nearshore wave model(SWAN) in an ocean wave model (WAM). It allows users to run the WAMmodel on a coarse grid (deep or intermediate water depth) and to run theSWAN model on a nested fine grid (shallow water) to achieve computationalefficiency. The system was used to hindcast waves in February 1993 along theBelgian and Dutch coasts. Model results from SWAN and WAM areintercompared and are also compared with buoy measurements. Goodagreement was found between WAM, SWAN and observations. In addition, anumerical investigation of depth refraction of waves was made for a uniform1:1100 sloping coast. A wave directional resolution of the order of 15° wasfound to be sufficient for SWAN to reproduce the wave refraction given bySnell's law while for WAM a directional resolution of 5" was required toachieve the same accuracy.

1 Introduction

Increased marine activities and coastal development have created an urgentneed for improved knowledge of sea-state condition in nearshore regions. Animportant tool to supply this knowledge is wave modelling. One of the mostadvanced wave models is the third generation WAM model (WAMDIG [7]),which has demonstrated excellent performance on global and regional scales inthe deep sea and in intermediate water depths. However, in nearshore areas, afine resolution (in geographic space) is required to account for the complexityof bathymetry and for the complicated physical processes involved. A wave

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254 Computer Modelling of Seas HI

directional resolution of the order of 5 ° is needed for WAM to reproduce thewave refraction given by Snell's law (Hubbert & Wolf [2]). The propagationtime step in the WAM model is limited by the Courant-Friedrich-Lewy (CFL)stability criterion, and the source term time step must be shorter than thepropagation time step. For high resolution applications the time steps mighteven become much smaller than the time scale of the physical processes

involved. The computation cost grows as mxn* when the wave directionresolution increases m times and the geographic space resolution increases ntimes. Moreover, for the WAM model, the propagation time step has to befurther reduced in order to fulfil a stability condition in wave direction space incases of strong depth gradient in nearshore regions. The major problem forroutine use of the WAM model in coastal regions is its prohibitively hugecomputational cost.

Recently, a wave model called SWAN (Simulation of WAves in Nearshoreareas) has been developed by Delft University of Technology (Ris et al. [6]).This model is also a third generation fully spectral model, solves the waveaction density transport equation without a priori spectral constraints, andtakes account of depth-induced wave breaking and triad interactions (which arenot included in the WAM model but important for wave evolution in nearshoreareas). The most attractive aspect of the SWAN model is its computationalefficiency since the model uses an implicit scheme allowing the use of a largetime step (greater than the CFL limit).

The objective of the present study was to develop a numerical scheme tonest the SWAN model into the WAM model, i.e., to allow users to run theWAM model on a coarse grid (deep or intermediate water depth) and to runthe SWAN model on a nested fine grid (high resolution) to achievecomputational efficiency. This paper is organised as follows. In section 2, abrief description of the WAM and SWAN models is given. The technicaldescription of grid nesting can be found in section 3. In section 4, we presentan application to the southern North Sea. A numerical investigation of depthrefraction of the WAM and SWAN models is presented in section 5, andsection 6 gives a summary, discussion and conclusions.

2. Model description

2.1 The WAM model

The WAM (Cycle 4) model is a third generation wave model which solves thewave transport equation explicitly without any ad hoc assumption on the shapeof the wave spectrum. The basic equation in Cartesian co-ordinates is

+A( )+A(,F)+cA(c A/•> . /-» \ .r / ^ \ y / — \ c / /«\/-\ \ o / totot ox dy oa a oG


Computer Modelling of Seas HI 255

where F(cr,9) is the wave energy spectrum, t is the time, a is the intrinsic

angular frequency, 6 is the wave direction measured clockwise from truenorth, c,, c , c^ and c@ are the rate of change of the position and propagation

direction of a wave packet travelling along a path. The left hand side of theabove equation represents the local rate of change of wave energy density,propagation, shifting of frequency due to time variation in depth, andrefraction. The right hand side represents all effects of generation anddissipation of the waves, including wind input 5% , whitecapping dissipation

S , non-linear quadruplet wave-wave interactions S and bottom friction

dissipation Sy . A detailed description of the WAM model can be found in

Gtinther et al. [1] and Komen et al. [3].The WAM (Cycle 4) model computes the evolution of the wave energy

spectrum. The model can run for deep and shallow water conditions, caninclude depth and current refraction (steady depth and current field only), andcan be set up for any local or global grid with a prescribed topographic dataset.The model solves the wave propagation equation using the first order upwindexplicit scheme, therefore, the propagation time step is limited by the CFLcondition. The propagation can be computed on a spherical or a Cartesian grid.The source term integration is solved by a semi-implicit scheme. Nesting ofgrids is possible for WAM. The boundaries of a nested fine grid have to be pre-defined in the set-up of a coarse grid run. The corner points of the nested finegrid have to coincide with nodes of the coarse grid. The coarse and fine gridmodel runs are separate. At the interface boundary, the wave spectra are outputat the boundary points of the coarse grid from the coarse grid run, and theninterpolated in space and tine to the boundary points of the fine grid. They areconsequently used as boundary condition for the nested fine grid run (one-waynesting).

2.2 The SWAN model

The SWAN (Simulation of WAves in Nearshore areas) model is based on theaction balance equation since the wave energy density is not conserved inpresence of currents, whereas action density is conserved. The equation solvedby the SWAN model is

where Af(a,9) is the wave action density (=F(a,9)/a), 0 is the wavedirection measured counterclockwise from the positive x-axis of the problemco-ordinate system. The source terms in SWAN include the wave energygrowth by wind input, wave energy transfer due to wave-wave non-linearinteractions (both quadruplets and triads), the decay of wave energy due to



whitecapping, bottom friction and depth-induced wave breaking. A detaileddescription of the SWAN (Cycle 2) model can be found in Ris et al. [6].

The basic spectrum considered in SWAN (Cycle 2) is the action densityspectrum. The model propagates the wave action density of all components ofthe spectrum across the computational area using an implicit scheme. It has thegreat advantage that the propagation time step is not limited by any numericalcondition since the scheme is unconditionally stable in geographic and spectralspace. Therefore, for high resolution applications in shallow water, thecomputation is expected to be more economic than WAM since a longer timestep can be used in SWAN. In geographic space the scheme is first-orderupwind and it is applied to each of the four directional quadrants of wavepropagation in sequence (i.e., divided into four sweeps). In the spectral spacethe scheme is variable between an upwind scheme and a central scheme. Thepropagation is done in a Cartesian co-ordinate system i.e., in a flat plane andnot on a spherical earth. The numerical scheme used for the source termintegration can be chosen by users between the fully implicit, semi-implicit orexplicit scheme. SWAN can be run in several modes: first-, second- or third-generation mode, according to the chosen level of parameterization. Nesting ofgrids is possible in SWAN. The idea of nesting in SWAN is similar to that inWAM. However, the corner points of the nested fine grid in SWAN do nothave to coincide with nodes of the coarse grid.

3. Technical description of grid nesting

Since the SWAN user interface is through a pre-defined command file, a newcommand had to be developed to allow the nested fine grid run using SWAN.A command called BOU WAMN has been developed in the present study; adetailed description can be found in Appendix A.

To accomplish the function of the new command in SWAN, five routineswere developed in the pre-program SWANPRE of the SWAN model. Theseroutines are designed to• read the wave spectra information from the output files of the WAM model.• convert the wave direction from WAM to the wave direction convention of

the SWAN model by the relationship: 6^^ = 90° -8%% , where 0^^ isthe wave direction in SWAN and 9 %%^ is the wave direction in WAM.

• convert the wave energy spectrum to the wave action density spectrum by

the relationship: jV(a,9) = — ILL21.27i a

• interpolate the wave spectra linearly in frequency and direction space, ingeographic space and in time.

• find the corresponding point index (KWAN •> /%*%*) in SWAN for a point) in WAM by the following equations:



^ SWAN -mxc + \, !¥„ylenw

myc +xlenw

where xlon and ylatwAM are the geographic location of the WAM point inlongitude and latitude direction (in degree), respectively; xpc and ypc are thegeographic location of the most south-west corner of the computational grid inSWAN in longitude and latitude direction (in degree), respectively; mxc andmyc are the numbers of meshes of the nested grid along the x and y axes,respectively, defined in the SWAN command CGRID; xlenw andylenw are thelengths of the computational nested grid in SWAN in longitude and latitudedirection (in degree), respectively.

4. Application to the southern North Sea

Using the grid nesting technique described above, a hindcast study for theperiod of February 1993 was carried out in the southern North Sea, where thebathymetry is shallow and complicated. A severe storm occurred on the 21st ofFebruary. The model was implemented on a nested grid. To account for theswell generated in the Norwegian Sea and propagating into the southern NorthSea, a WAM coarse grid covers the whole North Sea from 48° TV to 70° jVlatitude and from TW to 12° E longitude on a grid of 25 x 48 points, with aspatial resolution of about 50 km. Close to the Dutch and Flemish coasts, a finegrid with a spatial resolution of about Whn was set up, nested into thecoarse grid. Figure 1 shows the nested fine grid model bathymetry and exact

Fig. 1 The bathymetry of the southern North sea and locations of the twoBelgian buoy stations.


Computer Modelling of Seas HI

locations of two Belgian buoy stations.We use wind fields of the United Kingdom Meteorological Office (UKMO)

atmospheric model as the model forcing. Analysed fields or forecasts are useddepending on their availability. They are interpolated to the WAM coarse grid.The work of Ovidio et al. [5] showed that this wind forcing used to drive thewave model is of good quality.

The WAM coarse grid model was run for the whole month of February1993. The nested fine grid was run for the storm period from 18 to 23 Februaryusing both WAM and SWAN. Note that the same wave physical processes(wave growth and decay) in the SWAN runs were selected as in WAM. Thedirectional resolution of 15° was used in both models. The time steps used are5 and 30 minutes for WAM and SWAN, respectively. It is found that the totalcomputation (CPU) time used by SWAN is half that required by WAM.Figures 2 (a) and (b) show time series of the hindcast significant wave heightby WAM and SWAN and wave buoy observations at A2 Buoy (A2B) andWesthinder (WEH), respectively, for the storm period from 18 to 23 February1993. Very good agreement is found between the model results and wavemeasurements. The WAM model performs better than the SWAN model,which underestimates the wave height.

(a) A2B (b) WEH

2/18/93 2/19/93 2/20/93 2/21/93 2/22/93 2/23/9 2/18/93 2/19/93 2/20/93 2/21/93 2/22/93 2/23/9date (m/d/y) date (m/d/y)

Fig. 2 Time series of the significant wave height from WAM, SWAN andobservations: (a) at A2 Buoy (A2B); (b) at Westhinder (WEH).

In order to check if the nesting procedure is correct, the significant waveheight and the mean wave direction from WAM and SWAN were output every3 hours for the whole nested grid domain. As an example, Figure 3 shows thehindcast wave height and mean wave direction from WAM and SWAN,respectively, at 0600 GMT, 21 February, 1993. It is concluded that the nestingprocedure is successful and correct since the wave height and direction at openboundaries are identical for WAM and SWAN. Inside the nested grid domain,the significant wave height values from SWAN are generally smaller thanthose from WAM. Since we used the same physical processes in both models,it seems that the propagation scheme in SWAN is more dissipative than that in



Fig. 3 The hindcast significant wave height and mean wave direction at 0600GMT, 21 February, 1993: top for WAM; below for SWAN.



WAM. A relatively large discrepancy in mean wave direction (about 40°) isvisible in the Thames Estuary South of Harwich, UK. The discrepancy wasinvestigated by switching off all source terms and re-running the models. Wefound that in this area, changes in the mean wave direction with and withoutthe source terms are small for SWAN, but large for WAM. The wave meandirection from SWAN is insensitive to wind direction in less windy conditionsin the Thames Estuary since the propagation scheme is dissipative. In addition,an idealised test for depth refraction of waves was carried out in the followingsection.

5. Numerical investigation of depth refraction of the models

We consider a single frequency swell wave of approximately 10s period(0.0985Hz) approaching at 60° with respect to the isobaths from 80m to 10mwater depth for a uniform 1:1100 sloping coast for 4 days. Assume no currentspresent. Runs were carried out using SWAN and WAM with four differentdirectional resolutions, namely, 30°, 15\10\ 5°. The space resolution is about5.5 km. The time steps used are 3 and 30 minutes for WAM and SWAN,respectively. The mean wave direction values were output at pointscorresponding to water depth decreasing in 5 m intervals. The theoreticalsolution for wave refraction (depth only) is also obtained using the SnelFs law.Figures 4 (a) and (b) show the mean wave direction changes due to depth

(a) SWAN (b) WAM

40 20Water depth (m) Water depth (m)

Fig. 4 Mean wave direction changes due to depth refraction across a uniform

slope 1:1100 for direction resolution of 30% 15°, 10°, 5% with analyticsolution from the Snell's law: (a) SWAN; (b) WAM.

refraction across the slope. It is clear that increasing the directional resolutionin WAM and SWAN results in closer mean wave direction to the analyticalsolution from Snell's law. For WAM, a wave directional resolution of the orderof 5 * is needed to reproduce the wave refraction obtained with Snell's law, this



finding agrees with the conclusion of Hubbert & Wolf [2]. For SWAN, a wavedirectional resolution of the order of 75 ° is fine enough to produce results inreasonable agreement with SnelFs law. SWAN performs much better thanWAM in shallow water (water depth in order of 15 m).

Figure 5 shows the relative total CPU time as a function of the directionalresolution in WAM and SWAN. As the directional resolution is increased, ahigher computational cost is entailed. To reproduce the exact solution of waverefraction, SWAN requires less total CPU time than WAM. The CPU timerequired per time step in SWAN is much more than in WAM, but since thetime step in SWAN (30 minutes) is much longer than in WAM (3 minutes,limited by the CFL condition), on balance SWAN is faster. In terms of the total

CPU time, SWAN is expected to bemuch more efficient than WAM innearshore regions, where a higherspatial and directional resolution isrequired to define bathymetricfeatures accurately and to predictwave refraction correctly. On theother hand, in deep and intermediatewater depth, SWAN is not expected

Drectionai resolution (in degree) to be more efficient than WAM sincetime steps for WAM and SWAN can

Fig. 5 Relative total CPU time be nearly the same.

6. Discussion and conclusions

100 -r

U

The main difficulty for the use of the WAM model in high resolutionapplications in coastal regions is its prohibitively high computational cost. Thispaper has presented a numerical scheme to nest SWAN - a nearshore wavemodel - into WAM - an ocean wave model. This scheme allows users to run theWAM model in deep or intermediate water depth with a coarse grid and to runthe SWAN model in shallow water with a fine grid (nested into the coarse grid)to achieve the computational efficiency. An alternative approach to improvecomputational efficiency of the WAM model in shallow water is to develop ascheme for integrating the first-order upwind propagation numerics that splitsthe time step based on frequency. This permits the use of a longer time stepwhere possible and a substantial speed up of the WAM model (Luo et al. [4]).

The grid nesting technique was applied to a wave hindcast in Belgian andDutch coastal waters with a spatial solution of about 10km. Good agreementwas found between WAM, SWAN and observations. SWAN tends tounderestimate the significant wave height. This underestimation seems tosuggest that the numerical dissipation in SWAN is higher than in WAM. Anaccurate high-order propagation scheme is required to solve this problem.SWAN is 2 times more efficient than WAM. SWAN is expected to be much



more efficient than WAM at much finer resolutions (than 10km) since there isno limitation for the time step used in SWAN, while the time step in WAM islimited by the stability condition. However, the CPU time required per timestep by SWAN is more than WAM. Therefore, for deep and intermediatewater depth applications, SWAN is not expected to be more efficient thanWAM since the time steps for WAM and SWAN can be similar in magnitude.

Numerical tests were performed to assess the accuracy of depth refraction inSWAN and WAM for a uniform 1:1100 slope coast. A wave directionalresolution of the order of 75° is found to be fine enough for SWAN toreproduce the wave refraction given by Snell's law while a directionalresolution of 5° is required for WAM to achieve the same accuracy.

Acknowledgement

This work was partly supported by the Human Capital Mobility Program of theEuropean Commission (contract No ERB CHB GCT 920230) and by DelftUniversity of Technology. Discussions with L. Holthuijsen, N. Booij and R.Padilla-Hernandez and M. Sclavo are gratefully acknowledged. We also thankthe Afdeling Waterwegen Kust, Belgium for providing the buoy data.

References

1. Glinther, H., Hasselmann, S. & Janssen, P.A.E.M. The WAM modelCycle 4, Report No. 4, Hamburg, 1994.

2. Hubbert, K.P. & Wolf, J. : Numerical investigation of depth and currentrefraction of waves, J. Geophys. Res. 96 C2, 2737-2748, 1991.

3. Komen, G.J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S. &Janssen, P.A.E.M. Dynamics and Modelling of Ocean Waves, CambridgeUniversity Press, Cambridge, 532p, 1994.

4 Luo, W., Flather, R.A. & Monbaliu, J. An efficient computational schemefor the use of the third generation WAM model in coastal regions,submitted for publication in J. of Waterway, Port, Coastal, and OceanEngineering, 1997.

5. Ovidio F., Bidlot, J. -R., Van den Eynde, D., Luo, W. & Monbaliu J.Validation and improvement of the quality of the operational wave modelmu-WAVE by use of ERS-1, buoy data and the WAM forecast, Proceedingsof the First ERS-1 Pilot Project Workshop, Toledo, Spain, ESA SP-365,21-28, 1994.



6 Ris R.C., Booij, N., Holthuijsen, L.H. & Padilla-Hernandez, R. SWANCycle 2 User Manual - Simulation of WAves in the Nearshore Zone, DelftUniversity of Technology, the Netherlands, 1996.

7. WAMDI group : Hasselmann, S., Hasselmann, K., Bauer, E., Janssen,P.A.E.M., Komen, G.J., Bertotti, L., Lionello, P., Guillaume, A., Cardone,V.C., Greenwood, J.A., Reistad, M., Zambresky, L. & Ewing, J.A. TheWAM model - a third generation ocean wave prediction model, J. Phys.Oceanogr., 18, 1775-1810, 1988.

Appendix A

The new command developed in the present study is as follows.| \->CRAY |\UNF< >I \WK |

BOU WAMN 'fname'< >[mxw] [myw] [xlenw][ylenw][xpc][ypc]I I|FREE |

With the BOU WAMN command a nested SWAN run can be carried out withthe boundary conditions obtained from a coarse grid WAM run (WAM Cycle4). For this nested SWAN run the user has to give the CGRID command (seeSWAN User Manual, Ris et al., 1996) to define the computational grid beforethis boundary command. The computational grid for SWAN in geographicspace is the area bounded by the WAM nested boundary points. The range andresolution of frequency and direction of the WAM output will be overridden bythose in the CGRID command, i.e., interpolation will be made in frequency anddirection space in the SWAN model if the range and resolution of frequencyand direction of the WAM output are different from those defined by theCGRID command. In the above,'/name' name of a file that contains the names of all WAM files containing

the nested boundary conditions in time-sequence, which is definedby the parameter IDELRES in the WAM model, if IDELRES=24h,it means one file per day. For example, the contents of 'fname' canlook like:

CBO9212020000, CBO9212030000,..., etc.SWAN will read these WAM files. These files can be in an binaryform, or in a free format.

UNF the WAM files are binary unformatted file.CRAY input will be read from file created by the CRAY version ofWAM. Note that the WAM user should add an additional line to the script file.The command 'assign -F f77 -N ieee u:10', must be placed before theexecution statement in the script file. This converts FORTRAN unformatted


264 Computer Modelling of Seas III

I/O numeric data so that the SWAN model can read the unformatted filescreated by the CRAY.WK input will be read from file created by the workstation version of

WAM.FREE the WAM files are in a free format.mxw the number of meshes of the WAM coarse grid bounded the nested

fine grid in the longitude direction (one less than the number ofgrid points in the direction). The value of mxc defined in theSWAN command CGRID should be a multiple of mxw.

myw number of meshes of the WAM coarse grid bounded the nestedfine grid in the latitude direction (this number is one less than thenumber of grid points in the direction). The value of myc definedin the SWAN command CGRID should be a multiple of myw.

xlenw the length of the computational nested grid in SWAN in thelongitude direction (in degrees).

ylenw the length of the computational nested grid in SWAN in thelatitude direction (in degrees).

xpc the geographic location of the south-west corner of thecomputational grid in SWAN in longitude (in degrees).

ypc the geographic location of the south-west corner of thecomputational grid in SWAN in latitude (in degrees)


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