Wave spectral shapes in the coastal waters based on measured data off Karwar , west coast of 1 India 2

12 13 Understanding of the wave spectral shapes is of primary importance for the design of 14 marine facilities. In this paper, the wave spectra collected from January 2011 to December 2015 in 15 the coastal waters of the eastern Arabian Sea using the moored directional waverider buoy are 16 examined to know the temporal variations in the wave spectral shape. Over an annual cycle, for 17 31.15% of the time, peak frequency is between 0.08 and 0.10 Hz and the significant wave height is 18 also relatively high (~ 1.55 m) for waves in this class. The slope of the high-frequency tail of the 19 monthly average wave spectra is high during the Indian summer monsoon period (June-September) 20 compared to other months and it increases with increase in significant wave height. There is not 21 much interannual variation in slope for swell dominated spectra during the monsoon, while in the 22 non-monsoon period, when wind-seas have much influence, the slope varies significantly. Since the 23 exponent of the high-frequency part of the wave spectrum is within the range from -4 to -3 during 24 the monsoon period, Donelan spectrum shows better fit for the high-frequency part of the wave 25 spectra in monsoon months compared to other months. 26 27


Introduction 30 31
Information on wave spectral shapes are required for designing marine structures 32 (Chakrabarti, 2005)  An exponential curve y = kf b is fitted for high-frequency part of the spectrum and the 158 exponent (value of b) and the coefficient k is estimated for the best fitting curve based on statistical 159 measures such as least square error and bias. The slope of the high-frequency part of the wave 160 spectrum is represented by the exponent of the high-frequency tail. 161 162 For the present study, JONSWAP spectrum is tested by fitting for the whole frequency 163 range of the measured wave spectrum. It is found out that the JONSWAP spectra do not show a 164 good fit for higher frequency range, whereas Donelan spectrum shows better fit for the high-165 frequency range. Hence, JONSWAP spectrum is used for the lower frequency range up to spectral 166 peak and Donelan spectrum is used for the higher frequency range from the spectral peak for 167 single-peaked wave spectrum. Theoretical wave spectra are not fitted to the double-peaked wave 168 spectra. 169 water waves (where water depth is less than half the wavelength, d < L/2), this condition is not 176 satisfied during ~ 25% of the time due to waves with mean periods of 4.4 s or less. This study, 177 therefore, deals with shallow, intermediate and deepwater wave climatology. Hence, bathymetry 178 will significantly influence the wave characteristics. 179

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The persistent monsoon winds generate choppy seas with average wave heights of 2 m and 181 mean wave period of 6.5 s. Fig. 2 shows that in the monsoon, the observed waves had a maximum 182 H m0 of about 5 m, with H m0 of 2-2.5 m more common during this period. The maximum H m0 183 measured during the study period is on 21 June 2015 17:30 UTC (Fig. 2a). Mean wave periods 184 (T m02 ) at the measurement location ranged from 4-8 s (Fig. 2b). Wave direction during monsoon is 185 predominantly from the west due to refraction towards the coast. The fluctuation in H mo due to the 186 southwest monsoon is seen in all the years (Fig. 2a) whereas high waves (H m0 > 3m) have mean wave period in a narrow range (6.1-9.3 s) ( Table 2). 196 For waves with H m0 higher than 3 m, the Tp never exceeded 14.3 s and for waves with H m0 less 197 than 1 m, Tp up to 22.2 s are observed (Fig. 2c) and the long period swells (14-20 s) are with H m0 < 198 2.5 m. Around 7% of the time during 2011-2015, waves have peak period more than 16.7 s (Table  199 3). Peak frequencies between 0.08 and 0.10 Hz, equivalent to a peak wave period of 10 -12.5 s are 200 observed 31.15% of the time and the H m0 is also relatively high (~ 1.55 m) for waves in this class. 201 During the annual cycle, the wave climate is dominated by low (0.5 > H m0 >1m) intermediate-202 period (Tp ~10-16s) south-westerly swell. Waves from the northwest are with Tp less than 8 s (Fig.  203 3). 204

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The wave roses during 2011-2015 indicate that around 38% of the time during the period 206 2011 to 2015, the predominant wave direction is SSW (225) with long period (14 -18s) and 207 intermediate period (10 -14s) waves (Fig. 3). A small percentage of long-period waves having H m0 208 more than 1m are observed from the same direction in which more than 80% are swells (Fig. 3c). 209 Intermediate period waves observed having H m0 less than 1m, contain 20 -60% of swells. Around 210 10-15% of the waves observed during the period are from the west, which includes intermediate 211 and short period waves with H m0 varying from 1.5 to 3m. These intermediate period waves from 212 west having H m0 between 2.5 -3m contain more than 80% of swells. Waves from NW are short 213 period waves with H m0 between 0.5 and 1.5; in which swell percentage is very less showing the 214 influence of wind-sea (Fig. 3d). High waves observed in the study area consists of more than 80% 215 swells. 216

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Date versus year plots of significant wave height (Fig. 4) shows that H m0 has its maximum 218 values (H m0 >3m) during the monsoon period with a wave direction of WSW and peak wave period 219 of 10 -12s (intermediate period). The mean wave period shows its maximum values (6 -8s) during 220 the monsoon period. During January-May in all the years, H m0 is low (H m0 < 1m) with waves from 221 SW, W and NW directions. NW waves observed are the result of strong sea breezes existing during 222 this period. Both long-period (Tp > 14s), intermediate-period (10 < Tp <14s) and short-period (Tp 223 < 8s) waves are observed during this period and hence, the mean wave period observed is low 224 compared to the monsoon (Fig. 4d). During October to December, similar to the pre-monsoon 225 period, H m0 observed is less than 1m, but the wave direction is predominantly from SW and W, with 226 least NW waves. Short period waves are almost absent during this period, and the condition is 227 similar for all the years. The interannual variations in H m0 are less than 15% (Fig. 4). Primary 228 seasonal variability in waves is due to the monsoonal wind reversal. During January-March, there is 229 a shift in the occurrences of northwest swells. 230 231 4.2 Wave spectrum 232 233 The normalized wave spectral energy density contours are presented for different years to 234 know the wind-sea/swell predominance (Fig. 5). Normalisation of the wave spectrum is done to 235 know the spread of energy in different frequencies. Since the range of maximum spectral energy 236 density in a year is large (~ 60 m 2 /Hz), each wave spectrum is normalised through dividing the 237 spectral energy density by the maximum spectral energy density of that spectrum. The 238 predominance of both the wind-seas and swells are observed in the non-monsoon period, whereas 239 in the monsoon only swells are predominant (Fig. 5). The separation of swells and wind-seas 240 indicates that over an annual cycle, around 54% of the waves are swells. Glejin et al. (2012) 241 reported that the dominance of swells during monsoon is due to the fact that even though the wind 242 at the study region is strong during monsoon, the wind over the entire AS also will be strong and 243 when these swells are added to the wave system at the buoy location, the energy of the swell 244 increases (Donelan, 1987) and will result in dominance of swells. The spread of spectral energy to 245 higher frequencies (0.15 to 0.25 Hz) is predominant during January-May (Fig. 5) due to sea-breeze 246 in the pre-monsoon period (Neetu et al., 2006; Dora and Sanil Kumar, 2015). In the monsoon 247 during the wave growth period, the spectral peak shifts from 0.12-0.13 Hz to 0.07-0.09 Hz (lower 248 frequencies). 249 250 An interesting phenomenon is that the long-period (> 18 s) swells are present for 2.5% of 251 the time during the study period. The buoy location at 15 m water depth is exposed to waves from AS are the swells generated in the south Indian Ocean. In the monsoon season, the waves with 260 high-frequency are predominantly from west-southwest, whereas in the non-monsoon they are from 261 the northwest. In the non-monsoon period, the predominance of wind-seas and swells fluctuated 262 and hence the mean wave direction also changed frequently (Fig. 5). The average direction of 263 waves with H m0 < 1m shows the northwest wind-seas and the southwest swells, whereas, for high 264 waves (H m0 > 3m), the difference between the swell and wind-sea direction decreases. This is 265 because the high waves get aligned to the bottom contour before 15 m water depth on its approach 266 to the shallow water. 267

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The interannual changes of wave spectral energy density for different months in the period 269 2011-2015 are studied by computing the monthly average wave spectra for all the years (Fig. 6). In 270 the non-monsoon period, the wave spectra observed is double-peaked, indicating the presence of 271 wind-seas and swells, whereas during the monsoon, due to the strong southwest winds, single 272 peaked spectrum is observed, i.e. the swell peak with low-frequency and high spectral energy 273 density. Along the Indian coast, Harish and Baba (1986), Rao and Baba (1996) and Sanil Kumar et 274 al. (2003) found out that wave spectra are generally multipeaked and that the double peaked wave 275 spectra are more frequent during low-sea states (Sanil Kumar et al., 2004). Sanil Kumar et al. 276 (2014), Sanil Kumar and Anjali (2015) and Anjali and Sanil Kumar (2016) have also observed that 277 double-peaked spectrum in the monsoon period in the eastern AS are due to the locally generated 278 wind-seas and the south Indian Ocean swells. In the study area, from January to May and October 279 to December, the swell peak is between the frequencies 0.07 and 0.08 Hz (12.5 < Tp < 14.3s), but 280 in the monsoon period, the swell peak is around 0.10 Hz, in all the years studied. This shows long-281 period swells (Tp > 13s) in the non-monsoon period and intermediate period swells between Indian and Arabian Peninsula, with the highest wind speed of 215 km/h and affected the 300 areas of India, Pakistan and Oman. Significant interannual variation is observed in the wind-sea 301 peak frequency. Wave spectra averaged over each season (Fig. 7) shows that the interannual 302 variations in energy spectra averaged over full year period almost follows the pattern of wave 303 spectra averaged over monsoon period, indicating the strong influence of monsoon winds over the 304 wave energy spectra in the study area. Interannual variations within the spectrum are more for 305 wind-sea region compared to swell region. During the study period, the maximum spectral energy 306 observed is during 2011 monsoon. 307 308 For different frequencies, the monthly average wave direction is shown in Fig. 8. It is 309 observed that throughout the year the mean wave direction of the swell peak is southwest (200-310 250 o ). In the non-monsoon period, the wind-sea direction is northwest (280-300), except in 311 October and November. This is due to the wind-seas produced by sea breeze which has the 312 maximum intensity during the pre-monsoon season. Interannual variability in wave direction is 313 highest during October and November, where the wind-seas from southwest direction are also 314 observed. This is because, during these months, the wind speed and the strength of the monsoon 315 swell decreases, which makes the low energy wind-seas produced by the withdrawing monsoon 316 winds more visible. 317 318 Contour plots of spectral energy density (normalized) clearly show the predominance of 319 wind-seas and swells during the non-monsoon period (Fig. 9). Only Figs. 5 and 9 present the 320 normalised spectral energy density. In the monsoon period, the spectral energy density is mainly 321 confined to a narrow frequency range (0.07-0.14 Hz) and the wave spectra are mainly single peaked 322 with maximum energy within the frequency range 0.08-0.10 Hz, having direction 240 o . Glejin et al. 323 (2012) reported that in the monsoon season, the spectral peak is between 0.08 and 0. In the pre-monsoon period, wind-sea plays a major role in nearshore wave environment (Rao and 330 Baba, 1996). Wind-sea energy is found to be low during April 2015 (Fig. 6), because of reduction 331 in local winds. The occurrence of wind-seas is very less during most of the time in November 332 except during 2011, due to the deep depression ARB04. 333 . 334 The behavior of the high-frequency part of the spectrum is governed by the energy balance 335 of waves generated by the local wind fields. When the wind blows over a long fetch or for a long 336 time, the wave energy for a given frequency reaches the equilibrium range and the energy input 337 from the wind is balanced by energy loss to lower frequencies and by wave breaking (Torsethaugen 338 and Haver, 2004). The high-frequency tail slope of the monthly average wave spectrum in different 339 years shows that the slope is high (b< -3.1), during June to September and the case is same for all 340 the years studied (Table 4). During all other months, the exponent in the expression for the 341 frequency tail is within the range -3.1 to -1.5. The distribution of exponent values for different 342 significant wave height ranges shows that the slope increases (exponent decrease from -2.44 to - rate. These results are specific to breaking waves, but one might expect similar relations between 360 surface dynamics and dissipation rate for non breaking waves. A function of the form: A * exp( λ 361 H m0 ) + s0, with initial parameters of A = 8, λ = -2.4, s0 = -3.7 is found to fit the exponent of the 362 high-frequency tail data with the significant wave height (Fig. 11a). The functional representation 363 of the exponent of the high-frequency tail data with H m0 shown in Fig. 11a might be useful in 364 revealing the physical connection, and at the very least would provide a predictive basis relating 365 spectral slopes with mean significant wave heights as a basis for future research. It is shown in Fig.  366 11b that the exponent decreases (slope increases) as the mean wave period increases. The study 367 shows that the tail of the spectrum is influenced by the local wind conditions (Fig. 11c)  In the monsoon period, the spectrum is single peaked with high spectral energy density and 375 during this period JONSWAP spectrum is fitted up to the peak frequency and after that Donelan 376 spectrum is used. The monthly average wave spectra during the monsoon period for the year 2011, 377 is compared with JONSWAP and Donelan theoretical wave spectra in Figure 12. It is found that 378 JONSWAP and Donelan spectra with modified parameters describe well the wave spectra at low 379 frequencies and high frequencies respectively. The values for α and ϒ were varied from 0.0001 to 380 0.005 and 1.1 to 3.3 respectively to find out the values for which, the theoretical spectrum best fits 381 the measured spectrum and those values were used to plot the theoretical spectrum. The values of α 382 and ϒ thus obtained, for June, July, August and September are given in Table 6