WAVE PERIOD FORECASTING AND HINDCASTING INVESTIGATIONS FOR THE IMPROVEMENT OF

WAVE PERIOD FORECASTING AND HINDCASTING INVESTIGATIONS FOR THE IMPROVEMENT OF NUMERICAL MODELS Christian Schlamkow and Peter Fröhle University of Rostock/Coastal Engineering Group, Rostock Abstract: This paper describes comparative investigations which have been performed using the numerical wave model SWAN [1]. The comparisons bases on wave some hindcast simulation runs in the area of the western Baltic Sea, an area which is predominated by fetch-limited waves. The results of these simulations are compared to wave measurements that have been performed by directional wave buoys off the coast in water depths of approximately 12m. The comparisons are mainly focused on the wave periods and the shape of the directional wave energy spectra. The basis for the simulations are high-resoluted bathymetry informations and highresoluted wind informations. Different simulation variants with different formulations of physical effects (mainly wind-input terms) are used and the results are compared. As a result some recommendations for the application of these formulations are given. I. INTRODUCTION Waves are often the most important input parameter for design, dimensioning and construction of coastal structures. To get the necessary wave parameters, wave measurements, analytic wave forecast methods or numerical wave simulations are used. The state of the art wave forecast and hindcast method is the application of numerical wave models. The numerical model SWAN [1] is one of these models. SWAN is a so called community-model and widely used and accepted. SWAN is optimized for the simulation of nearshore waves and the simulations include some special physical effect in shallow water areas [2], e.g. triad interacting. In comparison to other wave hindcast methods the results of SWAN simulations show a good agreement to measured values [5,6]. Especially the wave hight and wave directions parameters are reproduced reasonably good. Unfortunately, the wave periods are often too short in comparison to measured wave periods. It is the aim of the presented investigations, to determine the reason for this effect. Different simulation runs which have considered different physical formulations have been performed. In a first approach this has been done by comparisons of measured and simulated wave parameters. In a second step, the shape of simulated directional wave energy spectra have also been compared to the measured wave spectra. identification of the effects of different physical formulations on the results, it is important that the other influences to the model accuracy are as small as possible. In many cases, the problem for the accuracy of wave models is the input data. In these investigations, high-resoluted bathymetry and wind input data are combined to a high-resoluted numerical model. The best available input data are used to ensure the quality of this part. A. Bathymetry The simulations area is located in the western part of the Baltic Sea. The used bathymetry data are a combination of different surveys and investigations [4]. The resolution of the bathymetry is x=1' and y=0.5' which is approximately 1km x 1km. Bathymetry and simulation area are shown in Figure 1. B. Wind The wind data is provided by the German Weather Service (DWD). The DWD operates a model system for weather forecast and hindcast purposes. The wind data for the wave model is computed by the so called Local Model (LM). The data bases on observations which are used for Re-analysing runs. The spatial resolution of the wind data is x= y 0.0625 what is approximately 7km x 7km. The time resolution of the wind input fields is one hour. II. DESCRIPTION OF MODEL SETUP Problems in the accuracy of numerical wave simulations can result from different reasons. In this investigation the problems of the SWAN model and the used physical formulations have been analysed. For the comparison of the results and the 493

Figure 1. Bathymetry and simulation area III. SIMULATION PARAMETERS The spatial resolution of the wave simulations is the same as the spatial resolution of the bathymetry, approximately 1km x 1km. Tests with higher spatial resolutions showed no significant differences for the wave parameters, especially of the waves calculated outside the breaker zone. For the calculation of wave parameters in very shallow water and, especially, close to a very divided coastline, it is obviously necessary to use higher resoluted bathymetries. The time resolution of the simulations is one hour, like the resolution of the wind input fields and measurements used for the comparison. The maximum number of iterations is limited to 20 (which had never been reached in our simulations) in order to get the same convergence to the measured data as for shorter time steps (e.g. 10 minutes) The wave energy spectrum (or, in SWAN action density spectrum ) is discretized from 0.02 Hz to 1 Hz, corresponding to the recommendations in the SWAN user manual [2] and to theoretical considerations about the wave periods appearing in the simulation area. The wave directions are discretized in 36 steps (10 ). The resolution in frequency-domain is calculated by SWAN (normally 41 meshes) in order to meet numerical demands [2]. All relevant physical effects (triads, friction, quadruplets if available) are activated for the simulations. Depth induced breaking was not activated for these simulations, because it does not occur in the considered water depths in the simulation area. A. Simulated variants SWAN can be used as a first, second or third generation wave model. This setting activates and/or deactivates different physical formulations in the simulation and can be combined with special switches for some special physical formulations. In general, the wave growth is described by two mechanisms: First: the waves grow linear in time caused by pressure waves in the wind field. Second: the waves grow exponential in time by transfer of wind energy in dependence to the wave energy, because higher waves cause bigger pressure differences in the wind field [3]. Based on these mechanisms the wind input energy term can be described as: in ( σ, θ) ( σ, θ) S = A+ BE (1) A and B are depending on the wave frequency, the wave direction and wind speed and -direction. The expression for A uses a formulation of Cavaleri and Malanotte-Rizzoli [7]. The term for B can be chosen according to Snyder et. al [8] or to Janssen [9]. A new non-linear saturation based whitecapping term combined with an exponential wind growth term by Yan [10] is provided by Westhuysen [11]. In GEN1-mode, the linear wind growth following a modified Cavaleri and Malanotte-Rizzoli approach [4] can be combined with the exponential wind growth description following a modified formulation developed by Snyder et al [4]. The whitecapping formulation of Holthuijsen and De Boer [12] is used in this mode. The consideration of quadruplet wavewave interaction is not possible. The GEN2-mode uses the same formulations as the GEN1-mode with the modification that the Phillips constant of the formulation is variable in this mode. The GEN3-mode is the state-of-the-art mode. In this mode, the (optional) linear wind growth term uses the Cavaleri and Malanotte-Rizzoli formulation [7]. The exponential wind growth uses the Snyder et al formulation [8]. Alternative the Janssen 494

[9] or Westhuysen [11] formulation can be used. The whitecapping formulations use the basic approach of Komen et al. [13] or alternatively Janssen [9]. In this mode, it is possible to consider quadruplet wavewave interactions, following the Hasselmann et al. [14] approach. The simulation variants and the corresponding switches and modes are complied in Table 1. TABLE I. DESCRIPTION OF SIMULATION VARIANTS V0 V1 V2 V3 V4 V5 V6 V7 IV. default GEN1-mode default GEN2-mode default GEN3-mode, linear wind growth term is not activated GEN3-mode with Janssen wind growth term, linear wind growth term is not activated GEN3-mode with Westhuysen wind growth term, linear wind growth term is not activated default GEN3-mode, linear wind growth term is activated with (default) proportionality coefficient of a=0.0015 GEN3-mode with Janssen wind growth term, linear wind growth term is activated with (default) proportionality coefficient of a=0.0015 GEN3-mode with Westhuysen wind growth term, linear wind growth term is activated with (default) proportionality coefficient of a=0.0015 WAVE MEASUREMENTS V. RESULTS Figure 3. Directional waverider buoy For a first assessment of the accuracy of the simulations in comparison to the measured values, the reduced wave parameters were compared. Bases on these results the shapes of the directional wave energy spectra have been compared. A. Wave parameters Figure 4 shows the time-series of measured and calculated significant wave heights (H m0 ). In this kind of diagram it is difficult to identify the problems of the accuracy of the simulations or e.g. systematic errors. Therefore scattering graphs are used. Figure 5 shows the comparison of measured and simulated (V2) wave heights. In this and following graphs, the measured values are plotted in the x-axis and the calculated values in the y-axis. Figure 2. Locations of wave measurements The wave measurements have been performed at different locations in the Baltic Sea (Figure 2). The measurements are covering different periods. For this paper, only one typical location is selected for the comparisons (Warnemünde, Figure 2). The water depth at the location of the buoy is d=12 m. It is planned to expand the analyses to the other showed locations indicated in figure 2. The used measurement-device is a Datawell directional waverider-buoy (Figure 3). The measurement device works on basis of acceleration measurements. The analyses of the raw data, wave energy spectrum, directional spectrum and wave parameters are available. Figure 4. Time-series of wave height at Warnemünde 495

Figure 5. Comparison of measured and calculated (V2) wave heights 1) Variants V0, V1 and V2 Figure 6 shows the graphs for the wave heights of the simulation-variants V0 (GEN1-mode), V1 (GEN2- mode) and V2 (default GEN3-mode). It is noticeable that in all three variants the most values with low and moderate wave heights show a good agreement to the measurements. Some outliers with calculated wave height of 0.1 m to 0.2 m and measured wave heights of 0.7 m to 1.3 m are caused by special conditions at the measurement location which was near the port of Rostock. In calm conditions sometimes ship waves (especially from ferry boats) are measured but obviously not simulated. In case of storm conditions and corresponding high waves, the results are differing. V0 underestimates these waves clearly. In variant V1 these values are also underestimated, but the deviation is not so high (Table 2). In variant V2 a small overestimation was observed, but these variant shows the highest accuracy of wave heights of all variants (Figure 6). The scattering graphs for the mean wave periods are shown in Figure 7. The calculated wave periods of the variants V0 and V1 are nearly the same. The values are underestimated in most cases. In variant V2 (default GEN3-mode) the wave periods are always clearly underestimated and not useable for many practical cases. The reason for this behaviour is explained in section 4.2 and is related to limited accuracy of the prognosis of the shape of the wave energy spectrum. The wave directions are compared in the scattering plot in Figure 8, where only wave directions of waves with a height exceeding Hm0=0.5m are accounted. In general, the simulated wave directions show a good agreement to the measured directions for all simulations. 2) Variants V2, V3 and V4 The simulation variants V2, V3 and V4 contain the three different wind input terms of the GEN3-mode in SWAN. Variant V2 uses the default formulation, V3 the Janssen formulation and V4 the Westhuysen formulation. Figure 9 shows scattering diagrams for the wave height of these three variants. The graph of variant V2 shown at Figure 6 and gives a good agreement to the measured values. The variant V3 (Janssen formulation) shows complete implausible wave heights. It seems that these problems depend on a bug in the actual implementtation of SWAN (version 40.51AB). The calculated wave heights of variant V4 are overestimated in many cases, especially in case of high waves and the accuracy of the wave heights is worse than in V2. The wave periods of V2, V3 and V4 are compared in the scattering graphs in Figure 10. Figure 10a is also known from Figure 7c. The wave periods, calculated in variant V3 are as implausible as the wave heights. The wave periods, calculated by variant V4 are the best of all evaluated simulations, but also to low in most cases. 3) Variants V5. V6 and V7 In the variants V5, V6 and V7 the exponential resp. non-linear growth wind input energy terms of Janssen, Komen and Westhuysen are combined with the linear growth terms of Cavalerie and Malanotte with the default proportionality term of 0.0015. Figure 11 shows the scatter graphs of simulated and measured wave heights for these variants. Variant V6 is again completely implausible (see above). Variant V5 and V7 show very similar results compared to the graphs in Figure 9. To analyze this, some further comparisons are made. Figure 12 shows results compared the wave heights of variant V2 to V5 and variant V4 compared to V7. It is obvious that these variants are identical in most cases. Hence, the variants V5, V6 and V7 are ignored for the further analyses 496

a) V0 b) V1 c) V2 Figure 6: Comparison of measured and calculated (V2) wave heights a) V0 b) V1 c) V2 Figure 7: Comparison of wave periods of V0, V1 and V2 a) V0 b) V1 c) V2 Figure 8: Comparison of wave directions of V0, V1 and V2

a) V2 b) V3 c) V4 Figure 9: Comparison of wave heights of V2, V3 and V4 a) V2 b) V3 c) V4 Figure 10: Comparison of wave periods of V2, V3 and V4 a) V5 b) V6 c) V7 Figure 11: Comparison of wave heights of V5, V6 and V7 498

a) V2 to V5 b) V4 to V7 Figure 12. Comparison of wave heights of V2 to V5 and V4 to V7 In Table 1 the slope of the best fit linear regression line = and the standard H m0, calculated a H m0, measured deviation of the deviations between calculated and measured values is printed. TABLE II. SLOPE OF BEST FIT STRAIGHT LINE AND STANDARD DEVIATIONS Variant Wave height Wave period slope of best fit straight line standard deviation of deviations [m] slope of best fit straight line standard deviation of deviations [m] V0 0.86 0.19 0.79 0.58 V1 0.96 0.17 0.81 0.60 V2 1.08 0.17 0.60 0.45 V4 1.16 0.19 0.78 0.56 Table 1 shows that the calculated wave heights are clearly underestimated in variant V0 and V1. Variant V2 overestimates the wave heights slightly, variant V4 overestimates the wave heights more. In all variants, the spread of the deviations between calculated and measured wave heights in nearly the same. The calculated wave periods are too short in every analysed variant. The smallest mean deviations were observed in variant V1 (but with the largest spread of deviations), the highest deviations shows V2. In summary, the experience of underestimated wave periods in SWAN is also verified in this study. B. Spectral wave informations The SWAN wave model is based on the calculation of the changes of the wave energy spectra. The above compared parameters are derived from the wave energy spectra. This means, these problems in the accuracy of these parameters has their reasons in problems of the simulation of the energy density spectra. In some of the investigated variants, the wave height is comparatively good reproduced, but the wave period is underestimated significantly. This indicates that the shape of the simulated spectra is not identical to the measured spectra. This is investigated in the next chapter. In the first part, the spectra of some selected time steps are compared. After that, average wave spectra are compared. 1) Comparison of results at selected time steps December, 4 th 1999, 00:00. On December, 4 th 1999, the highest wave height in the considered period occured. Table 3 shows the measured and calculated wave parameters for this time step. The maximum wind velocity was U=21 m/s (average over one hour) from West for the location at Warnemünde. TABLE III. WAVE PARAMETER AT 4 TH DECEMBER 1999, 00:00 Variant H m0 [m] T 02 [s] Measured 2.96 5.76 V0 1.91 4.55 V1 2.50 4.84 V2 3.05 3.87 V4 3.25 4.94 Figure 13 shows the comparison of the measured and the calculated wave energy spectra and illustrates possible problems of each variant: V0 underestimates the energy peak of the spectrum and also the high-frequency-tail (f>0.2 Hz). This results in a too low calculated wave height and in a too short wave period. In V1 is the energy at the peak-frequency also underestimated. The high- and low frequency-parts shows a fair agreement to 499

the measured values. The calculated wave height is too low. The wave period has the best fit in this time, but is still too low. The variant V2 shows the result of the default SWAN GEN3-mode. This variant results in a good agreement of the wave height. Unfortunately the wave period is the worst of all variants. The energy at the peak of the spectrum is too low, and the energy at the high-frequency-tail is overestimated. This results in small deviations of the wave heights, but strong underestimation of the wave periods. The basic shape of the spectrum shows the worst agreement to the measured values. In variant V4, the calculated wave height is too high. The reason for that is an overestimation in the low-frequency-tail of the spectrum, which results in to much wave energy in this part of the spectra. The other parts of the spectra show a comparatively good agreement with the measured values, also the peak of the spectrum. It must be pointed out, that the internal calculation runtime of SWAN computes wave periods shorter than an external program which uses the SWAN output spectra, because SWAN adds a analytical (diagnostic) high-frequency tail to the discrete spectrum. To assess the influence of this effect, the wave periods are computed using an external program. Table 4 shows a comparison of both wave periods (computed by SWAN and by external program). The analyses of the wave measurements are performed also without such a diagnostic tail, so that the parameter wave period has a different basis. This effect changes the wave periods about 12% (in variant V2). TABLE IV. Variant COMPARISON OF WAVE PERIODS, COMPUTED BY SWAN AND BY EXTERNAL PROGRAM T02 [s], computed by SWAN T02 [s], computed by external program V0 4.55 4.63 V1 4.84 4.91 V2 3.87 4.34 V4 4.94 5.28 January, 30 th 2000, 09:00 The event of January, 30 th 2000, 09:00 is characterized by wind from WNW with a maximum wind velocity of U=21.7 m/s (average over one hour). Table 5 shows the wave parameters measured and simulated for this time step. TABLE V. WAVE PARAMETER AT 30 TH JANUARY 2000, 09:00 Variant H m0 [m] T 02 [s] Measured 2.85 5.60 V0 2.05 4.67 V1 2.36 4.70 V2 2.96 3.93 V4 3.13 4.86 The comparison of measured and calculated spectra is shown in Figure 14. The shape of the spectra of variants V0, V1 and V2 is similar to the event at the 4 th December. The spectrum, computed by V4 shows the problem of this variant very clear: There is to much energy in the low-frequency-tail and also in the high-frequency-tail of the spectrum. This produce to high waves and to short wave periods. Figure 13. Wave energy spectra at 4 th December 1999, 00:00 Figure 14. Wave energy spectra at 30 th January 2000, 09:00 500

December, 26 th 1999, 04:00 At the December, 26 th the wind blows from South with maximum wind speed of U=6.3 m/s. At 04:00 was the local maximum in wind speed and wave height at location Warnemünde. Table 2 shows the measured and the calculated wave parameters for this time step. In this example the wave height is relatively low, but very good calculated by variant V2. Variant H m0 [m] T 02 [s] Measured 0.88 3.92 V0 0.66 2.82 V1 0.73 2.86 V2 0.89 2.19 V4 0.94 2.94 Table 6: Wave parameter at January, 26 th 2000, 04:00 The comparison of the spectra is shown in Figure 15. It is obvious that all calculated spectra overestimate the wave energy at the high frequencytail in this case. This results in too short wave periods. Furthermore, it is shown again, that the shape of the spectrum, calculated by variant V2, does not match the measured spectra. The wave energy is clearly overestimated at the high frequency-tail and clearly underestimated at the peak of the spectrum. Anyhow, the calculated wave high is nearly identical with the measured value. This is typical for the results of variant V2. The variant V4 shows the best match in the peak of the spectrum, but overestimates the wave energy at the low- and high-frequency-tail.this results in too high waves. January, 24 th 2000, 02:00 Variant H m0 [m] T 02 [s] SWAN T 02 [s] external program Measured 0.85 3.35 3.35 V0 0.79 2.98 3.08 V1 0.91 3.17 3.26 V2 0.88 2.39 3.82 V4 0.91 2.84 3.16 Table 7: Wave parameter at January, 24 th 2000, 02:00 The good agreement in the calculated wave periods (variants V1 / V4) is founded by good agreements of the shape of the spectra for the measured and calculated values. Figure 16 shows the comparison of the spectra for this event. It is obvious that the shape of the spectra from variant V4 shows a relatively good agreement, but the low frequency-tail is again overestimated. In this example another problem becomes apparent: The measured spectrum has no high frequency-tail, because the buoy only supply informations from f min =0.005 Hz to f max =0.635 Hz. It is possible that in cases of low waves the maximum frequency of natural waves is higher. This is not indicated by the buoy and may cause the problems in the agreement with simulated values. Figure 16: Wave energy spectra at 24 th January 2000, 02:00 Figure 15: Wave energy spectra at 30 th January 2000, 09:00 2) Comparison of averaged wave energy spectra In the previous paragraphs, measured and calculated spectra at discrete points in time are compared. To compare the spectra in general, average deviations from measured spectra are calculated. The measured spectra are approximated by linear interpolations to the discrete frequency sampling points of the SWAN calculated spectra and the energy of the measured spectra is subtracted from the calculated spectra and averaged over time. Figure 17 shows the result of the analyses. The graph shows that in variant V0 (GEN1-mode of 501

SWAN), the spectral peak is clearly underestimated. This results in too low wave heights. Variant V1 (GEN2-mode) has the same problem, but not so significant. The shape of the spectra calculated by variant V2 (default GEN3-mode), matches good around the peak of the spectrum, but the wave energy is overestimated in the high frequency-tail. That will lead to too short wave periods. The results of variant V4 (GEN3-mode with Westhuysen wind input term) clearly overestimates the energy at the low frequency-tail of the spectrum and also at the high frequency-tail. This tends to too high wave heights but relative low differences of measured and calculated wave periods. Figure 17: Average difference of calculated and measured wave energy spectra VI. CONCLUSION The performed investigations show, that wave forecasting and hindcasting with the SWAN-model are applicable for questions of coastal engineering. The results of such simulations have to be evaluated for the accuracy of the required wave parameters. SWAN simulations show a comparatively high accuracy for the wave heights, especially with the default GEN3-mode. Unfortunately, the wave periods have poor accuracy in this mode. The GEN3-mode with the Westhuysen wind energy input term shows a better agreement of the wave periods to the measured values in the Baltic Sea, an area where fetch limited sea-state predominates. It is recommended after these investigations to use the default GEN3-mode (Komen formulation) for questions concentrating to the wave heights and the GEN3-mode with Westhuysen formulation for questions concentrating to the wave periods. At this time it is not possible to give recommendations for both parameters in one mode. It is planned to enlarge the investigations described above to other locations. Also these investigations are a basis for improvements of the numerical model SWAN. It seems that these improvements should be concentrated to the change of the wind input energy term, because the change of this term causes significant changes in the shape of the calculated wave energy spectra. It seems that the formulation of Westhuysen resp. Yan is reproducing the shape of the wave spectra better than the other approaches. REFERENCES [1] Holthuijsen, L.H., Booij, N. & Ris, R.C. (1993): A spectral wave model for the coastal zone, Proceedings 2nd International Symposium on Ocean Wave Measurement and Analysis, New Orleans, Louisiana, July 25-28, New York, pp. 630-641. [2] The SWAN Team (2007): SWAN - Implementation manual. Delft University of Technology, Environmental Fluid Mechanics Section, available from http://www. uidmechanics.tudelft.nl/swan/index.htm (Version 40.51AB). [3] The SWAN Team (2007): SWAN - Technical documentation. 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