The change characteristics of the calculated wind wave fields near lateral boundaries with SWAN model

Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 DOI: 10.1007/s13131-016-0800-6 http://www.hyxb.org.cn E-mail: hyxbe@263.net The change characteristics of the calculated wind wave fields near lateral boundaries with SWAN model ZHANG Hongsheng 1 *, ZHAO Jiachen 1, LI Penghui 1, YUE Wenhan 1, WANG Zhenxiang 2 1 College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, China 2 Yangtze River Estuary Investigation Bureau of Hydrology and Water Resource, Yangtze River Water Resource Commission, Shanghai 200136, China Received 13 February 2015; accepted 15 April 2015 The Chinese Society of Oceanography and Springer-Verlag Berlin Heidelberg 2016 Abstract Since the wind wave model Simulating Waves Nearshore (SWAN) cannot effectively simulate the wave fields near the lateral boundaries, the change characteristics and the distortion ranges of calculated wave factors including wave heights, periods, directions, and lengths near the lateral boundaries of calculation domain are carefully studied in the case of different water depths and wind speeds respectively. The calculation results show that the effects of the variety of water depth and wind speed on the modeled different wave factors near the lateral boundaries are different. In the case of a certain wind speed, the greater the water depth is, the greater the distortion range is. In the case of a certain water depth, the distortion ranges defined by the relative errors of wave heights, periods, and lengths are different from those defined by the absolute errors of the corresponding wave factors. Moreover, the distortion ranges defined by the relative errors decrease with the increase of wind speed; whereas the distortion ranges defined by the absolute errors change a little with the variety of wind speed. The distortion range of wave direction decreases with the increase of wind speed. The calculated wave factors near the lateral boundaries with the SWAN model in the actual physical areas, such as Lake Taihu and Lake Dianshan considered in this study, are indeed distorted if the calculation domains are not enlarged on the basis of actual physical areas. Therefore, when SWAN is employed to calculate the wind wave fields near the shorelines of sea or inland lakes, the appropriate approaches must be adopted to reduce the calculation errors. Key words: SWAN model, wave factor, change characteristic, distortion range, water depth, wind speed, lateral boundary Citation: Zhang Hongsheng, Zhao Jiachen, Li Penghui, Yue Wenhan, Wang Zhenxiang. 2016. The change characteristics of the calculated wind wave fields near lateral boundaries with SWAN model. Acta Oceanologica Sinica, 35(1): 96 105, doi: 10.1007/s13131-016-0800-6 1 Introduction After the pioneering work of Gelci et al. (1956), several ocean wave models (e.g., The SWAMP Group, 1985; SWIM Group, 1985; The WAMDI Group, 1988; Tolman, 1991) have been developed. These models are denoted as the first, second and third generation wave models, depending on the level of parameterization of generation, dissipation and nonlinear wave-wave interactions (Tolman, 1991). On the basis of WAM wave model (The WAMDI Group, 1988), a third-generation spectral wave model (Simulating WAves Nearshore (SWAN)) has been developed to estimate wave conditions in small-scale, coastal regions by Booij et al. a- nd Ris et al. (1999). This model has been applied extensively in coastal regions (e.g., Xu et al., 2000; Lin et al., 2002; Signell et al., 2005;Shi et al., 2006; Rogers et al., 2007; Rusu et al., 2011; Gorrell et al., 2011; Zhang et al., 2013). It is also used to calculate the wind wave fields in inland lakes (e.g., Bottema and van Vledder, 2009; Moeini and Etemad-Shahidi, 2009; Zhang et al., 2015). The SWAN model assembles all relevant processes of generation, dissipation, and nonlinear wave-wave interactions. This model is a synthesis of state-of-the-art formulations, but many questions are still open, and significant improvements may be expected in the future (Ris et al., 1999). In fact, the SWAN model has been improved from different aspects by many researchers (e.g., Holthuijsen et al., 2003; Hsu et al., 2005; van der Westhuysen et al., 2007; Jia et al., 2010; Zijlema, 2010; Smith et al., 2011). Up till now, the latest version 41.01 has been released, and it is still in the process of improvement. In the SWAN model, it is presumed that all incoming wave energy is absorbed by the lateral boundaries. This leads the calculated wave heights near the lateral boundaries to be small. In fact, it is not certain for the wave energy near the lateral boundaries to be absorbed in practice. Therefore, the modeled wave factors near the lateral boundaries are often distorted. The lateral boundaries should be sufficiently far away from the area of interest to avoid the calculation error propagating into the area of interest (Booij et al., 1999). In order to obtain the right wind wave fields, the calculation domain should be larger than the area of interest. The focus of this paper is on studying the change characteristics of wind wave factors near the lateral boundaries and the distortion ranges in which the wind wave factors cannot be properly modeled, in the case of different water depths and wind speeds respectively. The SWAN model is briefly described In Sec- Foundation item: The National Natural Science Foundation of China under contract No. 51079082; the Natural Science Foundation of Shanghai City under contract No. 14ZR1419600; the Research Innovation Projects of 2013 Shanghai Postgraduate under contract No. 20131129; the Top Discipline Project of Shanghai Municipal Education Commission. *Corresponding author, E-mail: hszhang@shmtu.edu.cn

ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 97 tion 2. In Section 3, the basic parameters of SWAN are chosen, and the effect of spatial resolution on distortion range is discussed. The effects of water depth and wind speed on the change characteristics and distortion ranges of different wave factors near the lateral boundaries are studied in Sections 4 and 5, respectively. In Section 6, the SWAN model is employed to calculate the wind wave fields in Lake Taihu and Lake Dianshan. The two lakes, which are located in southeast China, belong to the Lake Taihu watershed, and their lateral boundaries are land. When the SWAN model is employed to simulate the wind wave fields of the two lakes, the characteristics of the calculated results near the lateral boundaries are discussed in detail. Finally, the conclusions are given is Section 6. 2 Description of SWAN The SWAN model is a famous third-generation wave model, which has been developed to compute random, short-crested waves in coastal regions with shallow water and ambient currents. It is a spectral wave model based on the action density balance equation (Booij et al., 1999; Ris et al., 1999). The SWAN model contains a number of physical processes which add or withdraw wave energy to or from the wave field. In Cartesian coordinates, the governing equation is expressed as (Booij et al., 1999): @N @t + @ @x c xn + @ @y c yn + @ @¾ c ¾N + @ @µ c µn = s ¾ ; (1) where N(σ, θ, x, y, t) is the wave action density as a function of intrinsic frequency σ, direction θ, horizontal coordinates x and y, and time t; c x, c y, c σ and c θ represent propagation velocity of action density in geographical x and y, frequency and direction space, respectively, and they are calculated on the basis of linear wave theory; S represents the source and sink terms, and S=S in +S ni +S ds, where S in represents the effects of wind wave generation, S ni represents the nonlinear triad and quadruplet wavewave interactions, S ds represents energy dissipation, including white capping, bottom friction, and wave breaking, respectively. 3 Basic case All of the calculated cases in the present paper were run with version 40.85 (The SWAN Team, 2013). The relevant parameters were set as: the directional resolution was 10, and the number of meshes was 36; the lowest and highest discrete frequencies were 0.08 and 1.0 Hz, respectively. When the above parameters were set, the common convention was referred. In the present study, the BSBT (Backward Space, Backward Time) scheme was chosen. The stationary mode was run when the wind fields are independent of time, whereas the non-stationary mode was run when the wind fields are dependent of time. The water areas in nearshore region were considered, and they are not very large. For example, the third longest river in the world the Yangtse River, is about 90.0 km at its broadest point, and most of the water depth is less than 10.0 m. Moreover, the areas of most of the calculation domains considered in the SWAN model are less than 100 km 100 km, and most of the water depths are less than 10.0 m (e.g., Booij et al., 1999; Bottema and Vledder, 2009; Ris et al., 1999; Rusu et al., 2011). Thus, in the present study, the lengths of calculation domain were set as 90.0 km in the x- and y-directions respectively, and the water depth h was set as 5.0 m. Both sides of the calculation domain in the y direction were set as the lateral boundaries, and the x axis was set along the centerline of calculation domain. The wind direction was assumed to match the positive x- axis direction, and it was 0. The wind speed at 10 m height above the still water level, U 10, was chosen as 10.0 m/s. In the x- and y-directions, three different grid sizes were set as 200.0, 500.0, and 1 000.0 m, and the corresponding mesh numbers were 450 450, 180 180, and 90 90 respectively. In Fig. 1, the calculated wave heights along the section of x=89.5 km, which have been developed fully, are shown in the case of different grid sizes. In this figure, H represents the significant wave heights. It is indicated that the modeled wave heights using the 500 m grid size matched those obtained using the 200 m grid size, but the modeled wave heights using the 1 000 m grid size differed near the lateral boundaries. In order to save computation time, the grid sizes were chosen as 500 m 500 m in the following cases considered. Fig. 1. Spatial distribution of the calculated wave heights along the section of x=89.5 km for different grid sizes. 4 Effects of water depth on distortion range The water depth of calculation domain varies. When the water depth increases, the fetch at which the wind wave is developed fully also increases. For example, for the case of U 10 =20 m/s, the wind wave is not developed fully when the fetch is shorter than 95.0 km if h is set as 10.0 m; whereas the wind wave has been developed fully when the fetch is 50.0 km if h is set as 6.0 m. Therefore, the length of calculation domain was assumed as 120.0 km in the x-direction when the effects of water depth on distortion range were studied. Correspondingly, the mesh numbers were set as 240 180. The water depths, h, were set as 1.0 m, 2.0 m,, 10.0 m, respectively. The wind speed and wind direction were set as 20 m/s and 0, respectively. The modeled results are shown in Fig. 2 for the case of h=10.0 m. In the figure, the horizontal coordinate is the dimensionless fetch, and the vertical coordinate is the dimensionless significant wave height. The dimensionless fetch ex and the dimensionless significant wave height eh are as follows: ex = gx U10 2 ; H e = gh U10 2 ; (2) Fig. 2. Spatial distribution of the calculated wave heights along the wind direction.

98 ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 where X is the fetch (m), g is the gravity acceleration (m/s 2 ), H is significant wave height (m). From Fig. 2, it is noticed that the wave heights are x-axis symmetrical and that the outgoing boundaries do not influence the calculated wave heights significantly. 4.1 Wave heights The distribution of modeled wave heights along the section of x=119.5 km for different water depths is shown in Fig. 3. In the present section, the modeled results in the middle of section of x=119.5 km are taken as the normal values. From Fig. 3, it can be seen that the modeled wave heights are different from each other in the case of different water depths, that the wave heights near the lateral boundaries are smaller than the normal values, and that the distortion ranges are also different from each other. In order to study the distortion range of wave heights near the lateral boundaries quantitatively, the distortion ranges are defined by the absolute and relative errors respectively. The modeled wave height at some point is considered to be invalid if the relative error at the point is greater than a certain value, such Fig. 3. Spatial distribution of the modeled wave heights in the case of different water depths. as 1%, 2%, or 3%; or the absolute error at the point is greater than a certain value, such as 0.02 m, 0.05 m, or 0.10 m. In this way, the relation of the distortion range of wave height to water depth is obtained, as shown in Fig. 4. In this figure, the vertical coordinate, D, represents the distance between the distorted point and the lateral boundary. From Fig. 4, it can be found that the distortion range of wave height near the lateral boundary increases as the water depth increases. The distortion range changes a little if the water depth is smaller than 4.0 m, and it increases obviously with the increase of water depth if the water depth is greater than 4.0 m. 4.2 Wave directions The modeled wave direction at some point is considered to be invalid if the absolute error at the point is greater than a certain value, such as 1, 2, or 3. In this way, the relation of the distortion range of wave direction to water depth is obtained, as shown in Fig. 5. From this figure, it can be found that the distortion range of wave direction obviously increases as the water depth increases. 4.3 Wave periods The distribution of modeled wave periods along the section of x=119.5 km for different water depths is shown in Fig. 6, where the vertical coordinate, T, represents the wave period. It is indicated that the modeled wave periods are different from each other in the case of different water depths, that the wave periods near the lateral boundaries are smaller than the normal values, and that the wave periods at the lateral boundaries are greater than the normal values. The modeled wave period at some point is considered to be invalid if the relative error at the point is greater than a certain value, such as 1%, 2%, or 3%; or the absolute error at the point is greater than a certain value, such as 0.05 s, or 0.10 s. In this way, the relation of the distortion range of wave period to Fig. 5. Curve of the distortion range of wave direction in the case of different water depths. Fig. 4. Curve of the distortion range defined by the relative errors of wave heights in the case of different water depths (a) and curve of the distortion range defined by the absolute errors of wave heights in the case of different water depths (b). Fig. 6. Spatial distribution of the modeled wave periods in the case of different water depths.

ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 99 water depth is obtained, as shown in Fig. 7. From this figure, it can be seen that the distortion range of wave period near the lateral boundary increases as the water depth increases. The differences among the distortion ranges are small if the water depth is smaller than 3.0 m or 4.0 m, and the distortion range increases obviously with the increase of water depth if the water depth is greater than 4.0 m. 4.4 Wave lengths The distribution of modeled wave lengths along the section of x=119.5 km for different water depths is shown in Fig. 8, where the vertical coordinate, L, represents the wave length. It is indicated that the modeled wave lengths are different from each other in the case of different water depths, that the wave lengths near the lateral boundaries are smaller than the normal values, and that the wave lengths at the lateral boundaries are greater than the normal values. The distortion range increases as the water depth increases. The modeled wave length at some point is considered to be invalid if the relative error at the point is greater than a certain value, such as 1%,2%, or 3%; or the absolute error at the point is greater than a certain value, such as 0.5 m or 1.0 m. In this way, the relation of the distortion range of wave length to water depth is obtained, as shown in Fig. 9. From Fig. 9, it can be seen that the distortion range increases obviously with the increase of water depth if the water depth is between 4.0 m and 8.0 m, and that the distortion range goes up and down,instead increases or decreases if the water depth is greater than 8.0 m. As mentioned above, the modeled wind wave factors near the lateral boundaries are obviously distorted; and the distortion ranges increase as the water depth increases. 5 Effects of wind speed on distortion range It is assumed that the water depth and wind direction are constant. They are set as 3 m and 0 respectively. The input wind speeds, U 10, are assumed as 7.0, 8.0,, 26.0 m/s, and the mean values of different Beaufort scales, including 6.7, 9.4, 12.3, 15.5, 22.3, and 26.5 m/s. The other parameters are set as those of Section 3. 5.1 Wave heights In the present section, the modeled results in the middle of section of x=89.5 km are taken as the normal values. The distribution of modeled wave heights along the section of x=89.5 km in the case of different wind speeds is shown in Fig. 10. From the figure, it can be seen that the modeled wave heights are different from each other in the case of different wind speeds, that the wave heights near the lateral boundaries are smaller than the normal values, and that the distortion ranges are also different from each other. In order to study the distortion range of wave heights quantitatively, the distortion ranges are defined by the absolute and relative errors respectively. The criterions for distortion are the same as Subsection 4.1. The relation of the distortion range of wave height to water depth is obtained, as shown in Fig. 11. From Fig. 11a, it is shown that the distortion range of wave Fig. 7. Curve of the distortion range defined by the relative errors of wave periods in the case of different water depths (a) and curve of the distortion range defined by the absolute errors of wave periods in the case of different water depths (b). Fig. 8. Spatial distribution of the calculated wave lengths in the case of different water depths. Fig. 9. Curve of the distortion range defined by the relative errors of wave lengths in the case of different water depths (a), and curve of the distortion range defined by the absolute errors of wave lengths in the case of different water depths (b).

100 ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 height near the lateral boundary decreases with the increase of wind speed if the wind speed is less than 20 m/s, and that the distortion range changes a little when the wind speed is greater than 20 m/s. From Fig. 11b, it is indicated that the distortion range of wave height increases when the wind speed is between 10 m/s and 20 m/s, and that the distortion range remains basically unchanged under the other circumstances. Compared Fig 11a with Fig. 11b, it is found that the variation trend of distortion range defined by the relative error is different from that defined by the absolute error. This is because the wave height increase as the wind speed increases, leading the relative error to be smaller. 5.2 Wave directions The criterions for distortion are the same as Subsection 4.2. The relation of the distortion range of wave direction to water depth is obtained, as shown in Fig. 12. From this figure, it can be found that the distortion range of wave direction near the lateral boundary decreases with the increase of wind speed, and that the decreasing trend of distortion range is weakened a little when the wind speed is greater than 20 m/s. 5.3 Wave periods The distribution of modeled wave periods along the section of x=89.5 km in the case of different wind speeds is shown in Fig. 13. The criterions for distortion are the same as Subsection 4.3. From Fig. 13, it can be seen that the modeled wave periods are different from each other in the case of different wind speeds, that the wave periods near the lateral boundaries are smaller than the normal values, and that the wave periods at the lateral boundaries are greater than the normal values. The relation of the distortion range of wave period to water depth is obtained, as shown in Fig. 14. Compared Fig. 14 a with Fig. 14 b, it is found that the variation trend of distortion range defined by the relative error is different from that defined by the absolute error. 5.4 Wave lengths The distribution of modeled wave lengths along the section of x=89.5 km in the case of different wind speeds is shown in Fig. 15. It is indicated that the modeled wave lengths are different from each other in the case of different wind speeds, that the wave lengths near the lateral boundaries are smaller than the normal values, and that the wave lengths at the lateral boundaries are greater than the normal values. The criterions for distortion are the same as Subsection 4.4. The relation of the distortion range of wave length to water depth is obtained, as shown in Fig. 16. From Fig. 16a, it can be seen that the distortion range decreases with the increase of wind speed. From Fig. 16b, it is indicated that the distortion range of wave height increases when the wind speed is between 10 and 20 m/s, and that the distortion range remains basically unchanged under the other circumstances. From Figs 15 and 16b, it is shown that the effect of wind speed on the distortion range is small. Compared Fig. 16a with Fig. 16b, it is indicated that the variation trend of distortion range defined by the relative error is different from Fig. 10. Spatial distribution of the calculated wave heights in the case of different wind speeds. Fig. 12. Curve of the distortion range of wave direction in the case of different wind speeds. Fig. 11. Curve of the distortion range defined by the relative errors of wave heights in the case of different wind speeds (a) and curve of the distortion range defined by the absolute errors of wave heights in the case of different wind speeds (b). Fig. 13. Spatial distribution of the calculated wave periods in the case of different wind speeds.

ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 101 Fig. 14. Curve of the distortion range defined by the relative errors of wave periods in the case of different wind speeds (a) and curve of the distortion range defined by the absolute errors of wave periods in the case of different wind speeds (b). Fig. 16. Curve of the distortion range defined by the relative errors of wave lengths in the case of different wind speeds (a) and curve of the distortion range defined by the absolute errors of wave lengths in the case of different wind speeds (b). on the results of Sections 3 to 5, the calculation domains for lakes should be properly enlarged based on the actual physical areas, when the wind wave fields are calculated with the SWAN model. Fig. 15. Spatial distribution of the calculated wave lengths in the case of different wind speeds. that defined by the absolute error. As mentioned above, the distortion ranges of wave factors, including wave height, wave period and wave length, defined by the relative errors are different from those defined by the absolute errors. For the cases considered, the distortion range defined by the absolute error remains basically unchanged and increases slightly when the wind speed is between 10 and 20 m/s, whereas the distortion ranges defined by the relative errors decrease as the wind speed increases. This is because the wave factors increase with the increase of wind speed, which leads the relative error to be smaller. The distortion range of wave direction near the lateral boundary decreases obviously with the increase of wind speed, and the decreasing trend is weakened a little when the wind speed is greater than 20 m/s. 6 Test cases The area of Lake Taihu is 2 427.8 km 2, and it is large. In contrast, Lake Dianshan has a small area of 63.0 km 2. The two lakes are shallow. For Lake Taihu and Lake Dianshan, the long-term average water depths are 1.89 m and 2.0 m, respectively. Based 6.1 Lake Taihu The bottom elevation (from the 1985 Chinese National Height Datum) and measurement stations are shown in Fig. 17. As shown in the figure, Lake Taihu is within a rectangle in which the horizontal coordinates of the southern, northern, eastern, and western boundaries are at y=3 422.00 km, y=3 491.00 km, x=488.60 km, and x=555.65 km, respectively. The lake lies between latitudes of 30 55 N and 31 34 N and longitudes of 119 53 E and 120 40 E (the Gauss-Kruger projection was employed to transform the geographic coordinate system into the Cartesian coordinate system). The wind fields and wave fields were measured at three stations: Dapukou, Gonghu, and Pingtaishan, hereafter named Stations T1, T2, and T3, respectively. The horizontal rectangular coordinates of Stations T1, T2, and T3 are (493.45 km, 3 460.54 km), (537.64 km, 3 480.32 km), and (510.06 km, 3 456.74 km), respectively. The measurement data from August 8 to September 2, 2007, were provided by the Hydrology and Water Resources Supervision and Measurement Bureau of the Taihu Basin Authority (Zhang et al., 2015). In this study, the measurement data on August 24 are used. The wind fields measured on that day are shown in Fig. 18. When the measured wind fields are used as the inputs, they are firstly converted into U 10. After modeling tests, it is proper for Lake Taihu considered that the grid sizes were chosen as 450 m in the west east direction (x-direction) and 500 m in the south north direction (y-direction). To define the calculation domain, three points, TA, TB, and TC (shown in Fig. 17), were selected. The selected three points were near the boundaries. The horizontal coordinates of points TA, TB, and TC were (520.71 km, 3 423.66 km), (515.83 km, 3 490.18 km), and (490.47 km, 3 456.91 km), respectively. The shortest distances between the shorelines and Point TA, TB, and TC were 263.98 m, 231.00 m, and 441.10 m, respectively. Three

102 ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 calculation domains were assumed, and the coordinates of their boundaries are listed in Table 1. This table shows that the four boundaries for Case I correspond to the boundaries of the rectangle shown in Fig. 17. In the figure, the rectangle just covers the outmost boundaries of Lake Taihu. In the eastern region, the waters are narrow and closed. Gales seldom impact the eastern levee. Thus, the eastern boundary of the calculation domains was not extended. The western, southern, and northern boundaries for Cases II and III were correspondingly extended by five meshes from the Cases I and II boundaries, respectively. Accordingly, grids of 149 138, 154 148, and 159 158 were adopted for Cases I, II, and III, respectively. Using the average wind fields of the three stations as inputs, the calculation results for Points TA, TB, and TC, and Station T2, which is close to the shoreline, on August 24 are shown in Fig. 19. From the figure, it can be seen that the calculated wave heights at Point TA for Case I were dif- Fig. 17. Sketch of the bottom elevation and measurement stations in Lake Taihu. Fig. 18. Measured wind fields on August 24, 2007 (20:00 o clock on August 24 is defined as the zero hour). ( ) for Station T1, (+) for Station T2, and (Δ) for Station T3. Fig. 19. Comparisons of the calculated wave heights for Lake Taihu on August 24, 2007, using different calculation domains. Solid line: Case I; dashed line: Case II; triangles: Case III. Table 1. Horizontal coordinates of boundaries of the three calculation domains for Lake Taihu and Lake Dianshan, respectively Lake Taihu Lake Dianshan Southern Northern Western Eastern Southern Northern Western Eastern Case boundary boundary boundary boundary boundary boundary boundary boundary y=3 422.00 y=3 491.00 x=488.60 x=555.65 y=3 439.40 y=3 452.50 x=299.96 x=310.96 y=3 419.5 y=3 493.50 x=486.35 x=555.65 y=3 438.90 y=3 453.00 x=299.46 x=311.46 y=3 417.00 y=3 496.00 x=484.10 x=555.65 y=3 438.40 y=3 453.00 x=299.46 x=311.46 Notes: In unit of km.

ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 103 ferent than those for Cases II and III, and the calculated wave heights at Points TB and TC, and Station T2, were equal for all three cases. The calculated wave heights at Stations T1 and T3 for all three cases were also compared and found to be equal. This is because Stations T1 and T3 are in the center region of the lake. Therefore, Case II was selected as the final calculation domain. This was because Case II was smaller than Case III, and so the computational time was not excessively wasted. 6.2 Lake Dianshan The bottom elevation chart and the three measurement stations are shown in Fig. 20. The Stations D1, D2, and D3 are located at (301.98 km, 3 442.48 km), (304.14 km, 3 444.13 km), and (306.39 km, 3 446.42 km), respectively. The wind fields measured on July 18, 2009 are shown in Fig. 21. Grid sizes were set to 100 m in the east west direction (x-direction) and in the north south direction (y-direction). To define the calculation domain, four points, A, B, C, and D (shown in Fig. 20), were selected; they were all near the boundaries. The horizontal coordinates of Points A, B, C, and D were (301.96 km, 3 440.00 km), (300.16 km, 3 441.20 km), (307.46 km, 3 452.30 km), and (310.76 km, 3 446.80 km), respectively. The shortest distances between the shorelines and Points A, B, C, and D were 600 m, 200 m, 100 m, and 100 m, respectively. Three calculation domains were assumed, and the coordinates of their boundaries are listed in Table 1. This table shows that the four boundaries for Case I correspond to the boundaries of the rectangle shown in Fig. 20. In the figure, the rectangle just covers the outmost boundaries of Lake Dianshan. The four boundaries of Case II were correspondingly extended five meshes from the Case I boundaries. Grids of 110 131 and 120 141 were adopted for Cases I and II, respectively. The calculation results for Points A, B, C, and D on July 18 are shown in Fig. 22. This figure shows that the calculated wave heights at Point A for Case I differed significantly from those for Case II; the calculated wave heights at Point B for Case I were somewhat different from those for Case II; and the calculated wave heights at Points C and D for Case I were the same as those for Case II. Therefore, the southern boundary of Case III was extended five meshes from the Case II boundary, with the other Case III boundaries being the same as the corresponding Case II boundaries. The calculated wave heights of four points for Case III are also shown in Fig. 22, and the data indicate that the calculation results for Case III were the same as those for Case II. Therefore, Case II was selected as the final calculation domain. Fig. 20. Sketch of the bottom elevation and measurement stations in Lake Dianshan. Fig. 21. Measured wind fields in Lake Dianshan. Solid line: Station D1; dashed line: Station D2; dash-dotted line: Station D3.

104 ZHANG Hongsheng et al. Acta Oceanol. Sin., 2016, Vol. 35, No. 1, P. 96 105 m. Therefore, if the water depth of calculation domain is greater than 4.0 m or so, the lateral boundaries should be sufficiently far away from the area of interest to avoid the calculation error propagating into the area of interest. When the water depth and wind direction are assumed to be constant, generally, the distortion ranges of the calculated wave factors decrease with the increase of wind speed. Moreover, for the same wave factor, the variation trend of the distortion range defined by the relative error is different from that defined by the absolute error. When the absolute error is used to define distortion range, the distortion ranges of the wave factors including wave height, wave period, and wave length change a little with the variation of wind speed; whereas the distortion range defined by the relative error decreases obviously with the increase of wind speed. This is because the calculated wave factors increase as the wind speed increases, which leads the relative error to be smaller. The distortion range of the modeled wave direction decreases as the wind speed increases. Whether the water depth or the wind speed is assumed to be constant, the modeled wave factors, including wave height, wave period, and wave length near the lateral boundaries are less than those along the x-axis; however, the modeled wave period and wave length at the lateral boundaries are greater than those along the x-axis. Therefore, when SWAN model is employed to calculate the wind wave fields, the calculation domain should be properly enlarged on the basis of the area of interest, Such as Lake Taihu and Lake Dianshan considered in this study, in order that the whole wind wave fields in the area of interest can be obtained accurately and the computation time is not wasted. The present study should also be useful to the knowledge and improvement of SWAN model. Fig. 22. Comparisons of the calculated wave heights for Lake Dianshan on July 18, 2009, using different calculation domains. Solid line: Case I; dashed line: Case II; triangles: Case III. 7 Conclusions The change characteristics of wind wave factors near the lateral boundaries of calculation domain, and the distortion ranges in which the wind wave factors cannot be properly modeled are studied in detail in the present paper when SWAN model is employed to model the wind wave fields. The cases with different water depths or different wind speeds are considered, the following summaries can be obtained. When the wind speed and wind direction are assumed to be constant, the distortion ranges of the calculated wave factors increase with the increase of water depth. The distortion range of wave height is small if the water depth is less than 4.0 m or so; whereas it increases obviously if the water depth is greater than 4.0 m or so. 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