Assessment of reliable wave climate at jetty on open coasts: role of 2D spatio-temporal spectrum

India has a very long coastline of about 7,500 km. Along the coastal belt, a wide range of industries (such as oil and refineries, ship building), infrastructures belonging to ports, harbours, nuclear power plants, naval establishments, fishing coastal communities, etc. are situated. Mumbai, being the financial capital of India, has witnessed rapid economic growth and an expansion of population in recent decades resulting in a massive increase in traffic, and thereby congestion. As Mumbai is situated on a peninsula on the west coast, there is a constraint in space for the further development of roads on the land. As such, to cope with the present and future demand of heavy traffic in coastal mega-cities like Mumbai, developing new transport links like a sea link bridge and coastal roads is essential. To determine safe operable and design wave conditions for the waterfront facilities in nearshore/coastal areas, reliable long-term historical information on ocean wave spectral parameters (such as significant wave height (H_s), peak wave period (T_p), mean wave period (T_m01), mean wave direction (θ_m), and directional spreading of wave energy) is very important. The underestimation of the operational wave climate for any waterfront facility may lead to an increase in downtime and strategic failure, whereas the overestimation of the wave climate will lead to non-utilisation of a facility and unnecessary investments in protecting structures.Therefore, it will affect the economic viability of the structure.

The most reliable information about ocean waves is obtained from in situ measurements. However, such measurements are seldom available for long-term duration along the Indian coast. Thus, a wave atlas prepared by the organisations of the National Institute of Oceanography (NIO) and the Indian Metrological Department (IMD) based on ships' observed data, remote-sensing data, reanalysed data, and numerical wave hindcast models are used to obtain information on the long-term historical wave climate. The wave atlas prepared based on the limited number of ships' observed data (on specific routes) has a limitation on its applicability along the entire Indian coast on a spatio-temporal scale. As the repeat cycle of satellite altimetry (remote-sensing data) for any location varies from ten to 35 days, the observational data obtained from the altimeter and scatterometer Synthetic Aperture Radar (SAR) do not provide continuous wave data for any location and also it has a tendency to miss observations for extreme events. On the other hand, the reanalysis datasets prepared by organisations like the European Centre for Medium-Range Weather Forecasts (ECMWF), National Centers for Environmental Prediction (NCEP) – National Center for Atmospheric Research (NCAR) can be used to hindcast the wave data. In reanalysed datasets with the help of data assimilation systems, the output obtained from a numerical weather prediction model is combined with the data received from satellites, aircraft, ships, Earth observation systems, etc. to improve the accuracy of the dataset. One of the recent-generation fine-resolution reanalysis wave datasets is available in ERA-5. Recently, the ERA-interim reanalysed database has been upgraded into ERA-5, wherein more observational data and output from high-resolution numerical models have been used during data assimilation to improve its accuracy in providing a wide range of variables such as wind, ocean wave, precipitation and evaporation. This dataset is available from 1940 to the present with spatial resolution ∼0.25 × 0.25° and temporal resolution ∼1 h. The performance of the ERA-5 dataset in predicting the in situ data was evaluated in the South China Sea (Liu et al. 2022) wherein the comparison of reanalysed data with the in situ dataset indicates that both compare well. As the ERA-5 datasets are available only at the specific grid points, in order to use the ERA-5 database to hindcast the wave climate in the nearshore areas wherein the reanalysed datasets are not available, it is essential to downscale the dataset with the help of numerical models, statistical methods, artificial neural networks, etc.

In recent decades, the state-of-the-art numerical models which have been most widely adopted by practitioners and scientific communities to develop global/regional models for the hindcast of the long/short-term historical spatio-temporal evolution of surface gravity waves (for operational, extreme conditions) in deep to intermediate water depths are the Wave Watch-III, Wave Action Model (WAM), Simulating Waves Nearshore (SWAN), third-generation (3G) phase average spectral wave model (like Mike-21 SW, TOMAWAC), etc. Depending on the wave climate obtained from the global, and regional wave models, the phase-resolving models based on the solutions of Boussinesq (Mike-21 BW) and mild slope equations (ARTEMIS) are adopted to obtain the nearshore wave conditions. Therefore, the global, regional models act as a wave atlas wherein the reanalysed historical datasets (wind, waves) are dynamically downscaled to the regional level so as to obtain the wind wave climate and are based on which short- to long-term strategic planning for various developmental activities in coastal and offshore areas (coastal zone managements, etc.) are carried out. As the global, and regional models cover a huge domain both on oceanic and coastal scales, the accuracy of these models should be ascertained with the help of in situ measurements before their practical application.

Information on the short-term and long-term wave climate for operational, extreme conditions over the Indian coast during the summer monsoon (south-west monsoon), north east monsoon and also during fair-weather seasons are of great importance for coastal communities, environmental/territorial planners and also for scientific communities. The analysis of short-term wave climate on the Indian coast during operational/extreme wave conditions has been analysed by many researchers with the help of in situ measured data, numerical models, etc. Based on the in situ measurements for a year, the surface wave dynamics of the Mumbai coast (north-eastern Arabian Sea) were analysed by Kumar et al. (2019) wherein with the help of partitioning of the wave spectrum, the seasonal variations of the dominance of seas and swell waves were studied. With the help of measured and modelled data, the spectral evolution during sea breeze and tropical cyclones were studied by Nair et al. (2021) wherein a Weather Research and Forecasting (WRF) model was used for the simulation of land–sea breeze and wind obtained from the ERA-5 database was used for the simulation of cyclonic waves. The impact of wave–tide interactions in modulations of wave height on the west coast of India (of Mumbai) was studied by Sirisha et al. (2023) wherein it was found that in order to improve the simulation accuracy, wave–tide interaction should be incorporated. Performance of ERA-5 wind data in the simulation of storm surge, wave–current interaction for Ockhi cyclone was evaluated by Samiksha et al. (2023) wherein it was found that a coupled hydrodynamic wave model forced with ERA-5 wind is able to simulate storm surge, wave climate, etc. The numerical model MIKE-21 SW forced with ERA-5 wind was used by Sirisha et al. (2017) to evaluate the accuracy of an operational weather forecast system at ESSO-INCOIS during extreme events and calm weather conditions in the north Indian Ocean.

The multi-scaled nested modelling approach wherein SWAN is nested in WW-III/WAM has been adopted by many researchers to forecast the wave climate for long-term duration over the Indian coast. Amrutha et al. (2016) and Sandhya et al. (2014) used SWAN nested in WW-III to hindcast waves in the nearshore areas (of Karwar on the west coast of India) and at the coast of Puducherry (east coast of India), respectively. Patra et al. (2019) applied SWAN nested in WAM to predict spectral wave characteristics over the Bay of Bengal on the east coast of India. Samiksha et al. (2021) evaluated the sensitivity of wave growth and bottom friction parameterisation schemes for the SWAN nested in WW-III to hindcast waves over the central west coast of India. MIKE-21, the third-generation spectral wave model, was used by Rajasree et al. (2022) to analyse wind wave climatological trends for annual and seasonal wave characteristics on the entire west coast of India. Sreelakshmi & Bhaskaran (2023) used ESTELA (Evaluating the Source and Travel time of the wave energy reaching a Local Area) method to investigate the source and travel time of wave energy to reach the Indian coast. It was observed that for all of the hindcast studies wherein the nested modelling approach was adopted, the entire Indian Ocean (IO) was modelled in WW-III/WAM/MIKE-21 and the global model was forced with the reanalysed wind (ECMWF) to simulate the complex wave system, and the main reason for considering the entire IO as the domain is to incorporate the influence of Southern Ocean swells in altering the local sea-swell regime on the Indian coast and thereby ascertain simulation of the multipeaked nature of the directional wave spectrum along the Indian coast wherein there is seasonal variation in various spectral parameters such as directional spreading, amount of peakedness in the spectrum, spectral peakedness parameter (γ), etc. As such, to hindcast the historical wave climate for a period of more than 20 years (the generally followed time-frame by the scientific community for assessing wave climate at any location), the computational cost will be enormous. Therefore, there is a need to optimize the computational cost by selecting an optimal domain and suitable forcing functions so that the realistic nature of wave climate in nearshore/offshore areas can be simulated at an optimal computational cost.

The availability of a spatio-temporally varying reanalysed 2D directional wave spectrum along the Indian coast (in deep or intermediate water depth) has motivated us to check its applicability in deriving a nearshore realistic wave climate wherein the computational cost can be minimised by considering a relatively small regional model domain (about 3,000–3,500 km²) rather than simulating the entire IO (about 5 × 10⁶ km²). The present study envisages the applicability of a spatio-temporally varying reanalysed (ERA-5) directional spectrum, reanalysed wind and TPXO tide as forcing functions for a coupled HD (hydrodynamic, Telemac-2D) and SW (spectral wave, TOMAWAC) model for simulation of a realistic historical wave climate of the Mahim Bay (at –6 m with respect to the chart datum (CD) of Bandra, Mumbai). This wave climate was analysed to obtain the downtime for the proposed extension of jetties to be built to transport heavy bridge components, construction materials, equipment, etc. required for Versova Bandra Sea Link (VBSL) bridge construction at Bandra. The proposed extension of jetties, being situated on the open coast, is directly exposed to the ocean waves as well as macro-tidal-level variation (maximum tidal range of about 5.0 m). As such, the wave climate of the Mahim Bay (6 m below CD) was derived by using coupled HD and SW models to simulate wave–current and wave–tide interactions so that modulation of wave spectral parameters due to the impact of tidal currents is incorporated. TOMAWAC, being a finite element model, with the help of unstructured triangular mesh, the complex coastal bathymetric gradient (Benoit et al. 1996) and irregular shape of the coastline are appropriately represented as compared with the structured grid finite difference models like WAM, WW-III and SWAN. Another advantage of using TOMAWAC over the nested model is that within a single domain, the deep-water and shallow-water processes (such as bottom friction, depth-induced breaking, and non-linear interaction between a triplet of waves) are simulated wherein over the offshore boundary of the domain a spatio-temporally varying 2D directional spectrum can also be applied as a forcing function. TOMAWAC has been used to hindcast long-term wave climate (23 years) along French coasts (Lafon & Benoit 2007) and to assess wave energy potential in the Bohai Sea and Yellow Sea (Dong et al. 2020); both the models were forced to reanalyse wind to obtain the historical wave climate. Lafon & Benoit (2007) carried out extreme value analysis on 23 years of hindcast wave climate to arrive at wave conditions for various return periods (50/100 years).

The importance of wave–tidal current interaction in modulating wave kinematics (frequency, wavelength, wave number) or wave dynamics (height, action conservation) was reported by Wolf & Prandle (1999). The field measurements indicated that for a water depth of less than 20 m, tidal current is impacted by the waves and strong (about 0.8 m/s) following tidal current (i.e. the directions of waves and tidal currents are the same with a tolerance limit of 20°) or opposing tidal current (i.e. the directions of waves and tidal currents are opposite with a tolerance limit of 20°), which modulates the wave kinematics and wave dynamics. Kumar & Kumar (2010) carried out wave–current interaction studies (at 14 m depth) in a macro-tide-dominated region with the help of field measurements wherein it was found that due to the strong opposing current (>2 m/s), wave directions were tuned with the current directions and the spectral width parameter also increased which resulted in the generation of a wider directional spectrum, whereas wave steepness decreased during the occurrence of a strong following current. Coupled 2D HD and 3G-SW model simulations have been carried out by many researchers to assess the wave and tidal current energy potentials in coastal areas wherein the impact of tidal current in modulation of wave spectral parameters has been discussed in detail. Coupled HD-SW modelling carried out by Beya (2020) and Beya et al. (2021) to assess the impact of tidal current on wave energy resources indicates that a strong tidal current can modulate various spectral parameters such as γ (maximum by about 2.5), directional spreading (maximum by about 40°), H_s (by about 6%), T_m01 (maximum by about 5 s), θ_m (maximum by about 5%), etc. Guillou (2017) described the necessity of integrating tidal forcing in wave modelling wherein it was found that a strong tidal current (about 3.5 m/s) induced wave refraction and that wave breaking will lead to significant (30%) semi-diurnal modulation of H_s. The impact of a strong tidal current (3 m/s) on a bimodal seas-swell wave system was studied by Halsne et al. (2024) where it was found that opposing currents can increase H_s and γ by about 45% and 160% respectively. Dynamically coupled wave–tide interaction modelling was carried out by Lewis et al. (2019) to assess the impact of tidal dynamics in modulating wave height. The study reveals that due to the strong tidal current (about 1 m/s) induced Doppler shift, the refraction pattern of waves gets modified and also there is an increase in H_s even during low tide levels. Studies carried out by Ho et al. (2023) based on field observation and numerical solutions reveal that depending on the frequency of waves, water depth and magnitude of tidal current, when the speed of tidal wave exceeds the celerity of wind waves, even the following tidal current may increase H_s by 25% and may also shift the peak period by seconds. Numerical modelling carried out by Nurfitri et al. (2018) on the wave–current interaction reveals that when there was an occurrence of following tidal current (0.8 m/s), there was a decrease in significant wave height by about 9.3%, while the opposing current increased wave height up to about 9.1%. As such, in the present study a coupled HD-SW model was used to assess reliable wave climate in macro-tide-dominated regions and the accuracy of the simulated wave climate obtained from the coupled model was assessed by comparing with the measured H_s and T_m01. Hindcast wave climate provides information about the predominant wave climate which was further propagated to obtain the wave conditions near the proposed extension of jetties by using a phase-resolving model (ARTEMIS). The details of the study area, data used and the results are discussed in the following sections.

The wave climate of the Mahim Bay (6 m below CD) was derived by using a coupled hydrodynamic (HD) (Telemac-2D) and wave model (TOMAWAC). The reanalysed dataset (wind, wave database) of ECMWF was used to arrive at the wave climate at 6 m depth of the Mahim Bay. As the proposed jetties are expected to be utilised for short-term duration (up to the year 2028), the timeframe used for analysing the wave climate is decided based on the generally adopted rule of thumb wherein it is considered that the time span used for analysing wave climate should be at least one-third of the desired return period of wave climate (Goda 2000; Lafon & Benoit 2007; De Leo et al. 2022). As such, the predominant wave climate was derived based on the past couple of decades of wave climate (1997–2018) wherein not only would the selection of the timeframe as the recent past couple of decades provide information about the recent predominant wave climate but also the impact of the climatological trend on the wave condition would be incorporated so that its impact could be considered while deriving the predominant wave conditions. Based on the predominant wave climate, at −6 m CD (the outer boundary of the ARTEMIS model), wave conditions at the proposed extension of jetties were determined by using the ARTEMIS model.

During the two-way coupling of Telemac-2D-TOMAWAC, the Telemac-2D runs first and the hydrodynamic simulation is carried out to incorporate the impact of major forcing functions like the harmonic constituents of tides (M2, S2, K2, N2, K1, O1, P1, Q1, M4, MS4, MN4, etc.) as obtained from TPXO (Egbert & Erofeeva 2002) and spatio-temporally varying ERA-5 wind. Depending on the wind speed obtained from the ERA-5 database, the drag force on the water surface was computed by Flather's formulation. The information on the simulated current, water level and wind data is then shared with TOMAWAC for the prescribed coupling time-steps. Based on the input received from the HD model and also depending on the spatio-temporally varying 2D spectrum applied on the offshore boundary of the TOMAWAC model, various spectral parameters (H_s, mean period, peak wave period, mean wave direction), wave-induced radiation stress, etc. are computed in TOMAWAC. Based on the computed radiation stresses, the hydrodynamic simulation is updated and the updated hydrodynamic condition is used for the TOMAWAC simulation in the next time-step. Thus, two-way coupling of Telemac-2D and TOMAWAC represents the interaction of true wind wave characteristics not only in deep water but also in nearshore areas wherein the impact of wave interaction, wave–current interaction, wave-tide interaction and other source/sink terms are incorporated.

TOMAWAC, being a phase average spectral wave model, in order to incorporate the impact of wave diffraction if the available Holthujisen's (2003) formulations (Awk 2018) are adopted in TOMAWAC, it has been observed that near the areas wherein the diffraction effect needs to be simulated for rapidly varying bathymetry, selection of very fine mesh resolution (decided based on the peak period of the directional wave spectrum) results in undesirable effects such as numerical noise, unreasonable local amplification of the wave characteristics, underestimation of wave climate, etc. Therefore, it can be inferred that the modification in the wavelength (Grey et al. 2010) and wave height caused due to the interaction of reflected and diffracted waves with the incident wave (near any obstacle) are not adequately simulated in TOMAWAC. In order to overcome this limitation of the phase average model, the phase-resolving model (based on the solution of the mild slope equation) is adopted. Studies carried out by Eikema et al. (2018) indicate that although the 2D phase-resolving mild slope model is computationally cheap as compared with the 3D non-hydrostatic model (like SWASH), it properly simulates the physical processes like wave reflection, diffraction, wave breaking, resonance of waves, wave damping due to bottom friction, etc., especially near structures, in the navigational channel, inside/in the vicinity of the harbour. A comparison of simulation of the TOMAWAC and ARTEMIS models (phase-resolving mild slope model) for the transformation of waves on the lee-side of a breakwater in Jukbyeon Port, Korea, was carried out by Do et al. (2022) wherein it was found that ARTEMIS showed better performance in simulating the wave transformation immediately on the lee-side of the breakwater as compared with TOMAWAC and it was also validated with the field measurement. In the present study, as the proposed jetties are situated immediately on the lee-side of the existing bund, in order to assess the wave condition near the jetties the phase-resolving model ARTEMIS was used to adequately incorporate the wave condition created due to the interaction between the incident waves and the refracted and diffracted waves.

A multi-scaled nested modelling approach was adopted in the present study wherein the regional model developed (by coupling Telemac-2D and TOMAWAC) was used to obtain the wave climate for a period of 22 years (00.00 UTC of 01.01.1997 to 23.00 UTC of 31.12.2018) at 6 m depth, i.e. on the offshore boundary of the ARTEMIS model. The domains of the regional and nested models are shown in Figure 2(a) and 2(b), respectively.

In the present study two approaches have been adopted to obtain the wave climate over the offshore boundary (P–Q) of the ARTEMIS model (Figure 2(a)) wherein in the first scenario (Scenario-I), the time series of spatio-temporally varying H_s, peak period and mean direction of a specific spectrum (JONSWAP with γ = 3.3) with constant angular distribution functions (such as by Goda (2000)) was applied over the offshore boundary of the regional model. In the second approach (Scenario-II), the spatio-temporally varying 2D directional spectra obtained from the ERA-5 database (with varying γ, directional spreading, amount of peakedness, etc.) were applied over the offshore boundary of the regional model. In order to ensure the appropriate propagation of spectral energy over the domain on a spatio-temporal scale, it is essential to appropriately select the directional/frequency discretisation components, frequency ratios, minimum frequency, etc. As such, the imposed directional spectra were divided into various sets of directional/frequency discretisation components (20/25; 25/25; 36/25, etc.) along with various frequency ratios (1.115, 1.11, 1.125, etc.), minimum frequencies (0.04, 0.05 Hz.), etc. It was found that 36 directional bins, 25 frequencies, the frequency ratio of 1.125, along with the minimum frequency of 0.05 Hz, provide the best simulation of the wave climate. As the directional bins were divided into 36/25 components of directional/frequency bins, the directional spectrum was divided into a total of 900 pairs of directional–frequency bins during each time-step of the simulation of the spectral wave model. The spatio-temporal interpolation of imposed spectra for 900 pairs of directional–frequency bins was carried out by modifying the existing subroutine ‘Manche’ of TOMAWAC. In order to calibrate the spectral wave model, Janssen's model was used for the generation of wind waves. Komen and Jansen's model, Battjes and Janssen's model, JONSWAP's model, the DIA method (Discrete Interaction Approximation) and LTA (Lumped Triad Approximation) model were used for the activation of the major sink terms such as white capping dissipation, depth-induced breaking, bed-friction-induced dissipation, non-linear quadruplet interactions, and non-linear transfers between triads, respectively. The selected values of depth-induced breaking co-efficients for Battjes and Janssen's model were α = 1, γ₁ = 0.8 and γ₂ = 0.78. The JONSWAP bottom friction coefficient was kept as G = 0.038 m².s⁻³. Chezy's bed roughness coefficient of 60 m²/s was applied over the domain for HD simulation and the K-epsilon turbulence model was selected for the HD simulation. The comparisons of measured and simulated H_s and mean wave period (for various scenarios) represented in time-series plots as well as in scatter plots are shown in Figure 3(c)–3(h).

The calibration plots indicate that the regional model forced with spatio-temporally varying 2D spectra provides better simulation of wave climate (with correlation coefficient, R = 0.92 and 0.83 for H_s and mean period, respectively) as compared with Scenario-I wherein time series of spatio-temporally varying H_s, peak period and mean direction of a JONSWAP spectrum (with constant peakdness parameter of 3.3) were applied over the offshore boundary. The bias of H_s indicates that the application of constant peakedness spectrum over the offshore boundary underestimates the wave height (by about 0.25–0.35 m) and period (by about 0.05–1 s). As the entire operation for the transportation of bridge spans, bridge components, etc. is to be carried out only during non-monsoon season, in order to ascertain zero downtime at berth, it is essential to obtain a reliable wave climate at the jetties. Therefore, in the present study, the simulation for the regional wave model was carried out for a period of 22 years wherein spatio-temporally varying 2D spectra with varying peakedness were applied over the offshore boundary to obtain the reliable wave climate of the Mahim Bay so that zero downtime at berth can be ascertained.

The proposed extensions of the jetties are expected to be utilised up to the year 2028 for the construction of the sea link bridge. The climatological trend of the wave conditions (H_s, T_m01) of the Mahim Bay region are assessed so that if there is an increase in the trend of H_s and T_m01 during the hindcast period of 22 years (1997–2018), its impact can be incorporated to assess the future wave conditions near the jetties. The plot of mean H_s and mean T_m01 during pre- and post-monsoon seasons for 22 years are shown in Figures 4(g) and 4(h), respectively.

The plots indicate that mean H_s during pre-monsoon is more than during post-monsoon season whereas there is no significant variation in the mean wave period for both seasons. The trend of H_s for the past 22 years indicates that there is a decrease in mean H_s through the years 1999–2004 and 1999–2005 for pre- and post-monsoon, respectively. The trend of mean H_s during pre- and post-monsoon seasons stabilises between the past maximum and minimum range for the years 2006–2018. The plot of change in mean T_m01 indicates that from 2001 to 2005 there is a decrease in the mean T_m01 and after that, it stabilises between the past maximum and minimum range. As such there is no evidence that there is an increase in the trend of wave conditions due to the change in climatological conditions. Therefore, the analysis of 22 years of data will provide sufficient information about the predominant wave climate for the extension of jetties wherein there is no necessity to extrapolate the historical wave climate to incorporate the impact of change in the climatological conditions. A similar type of trend in H_s was also reported by Rajasree et al. (2022) for the Mumbai region, wherein the entire IO was considered as a domain for the simulation of wave climate and the forcing mechanism for the SW model was ERA-5 wind. However, the said study has not discussed anything about the spectral characteristics of wave climate, i.e. multipeaked nature of spectrum, wave frequency, directional spreading of waves, γ, wave-tide interaction, etc. In the present study, details of spectral characteristics along with wave–tide interaction are simulated and discussed. The percentages of occurrence of wave climate (H_s and mean wave period) obtained from the regional model of the Mahim Bay (6 m depth) are shown in Tables 1 and 2, respectively.

The wave agitation model has three liquid boundaries (A, B and C) and one solid boundary (on the coast). The directional wave spectra obtained from the TOMAWAC model were applied over the liquid boundary ‘A’ and the other two liquid boundaries (B, C) were set as liquid boundaries with free exit of waves. The liquid boundary ‘A’, over which the wave spectrum was imposed, was selected in such a way that along the boundary there is negligible variation in the wave climate. The solid boundary was considered as a fully or partially reflecting type. The varying reflection coefficient (0.2–1.0) derived based on the slope of the coast/structure, depth available, and incident wave conditions (near the solid boundary), viz. surf similarity parameter (US Army Corps of Engineers Coastal Engineering Research Center 1984) were applied on the solid boundary. The reflection coefficient of 1.0 indicates that all incident wave energy will get reflected and is applicable for vertical structures. wherein the reflection coefficient of 0.2 indicates most of the incident wave energy will get absorbed/dissipated. The reflection coefficients specified for the existing bund and the vertical solid wall near the proposed jetties are 0.4 and 1, respectively. The predominant wave conditions near the proposed jetties are assessed by imposing a 2D wave spectrum over the liquid boundary of the ARTEMIS model rather than imposing a monochromatic wave with constant H_s, period and direction. During the propagation of the 2D spectrum, depending on the range of frequency/directional spreading, the imposed spectrum is discretised into a specified number of frequencies (20 numbers)/directions (25 numbers). Steady wave computation for each of these pairs (frequency, direction) is achieved by solving the mild slope equation associated with the prescribed boundary conditions. Finally, all the wave conditions achieved for the discretised frequency/direction are recombined to obtain the results for the propagation of random multidirectional waves. Thus, the imposition of a 2D spectrum over the offshore boundary of the ARTEMIS model enables the simulation of a realistic predominant wave climate rather than imposition of monochromatic waves wherein the impact of spectral peakedness parameter, angular distribution/spreading of wave energy, etc. are ignored.

The plots of distributions of H_s indicate that under the existing conditions, as the waves propagate from the two predominant directions (i.e. west, west-south-west directions) due to the presence of sufficient depth (about 4.0–6.0 m CD) in the wave propagation direction as well as due to the absence of submerged shoals, headlands, etc., the incident waves directly reach to the southern end of the proposed jetties (protruding beyond the existing bund) with negligible dissipation of the wave energy, whereas due to the presence of relatively shallower depths (about 0.1 m CD) around the headland towards the west-north-west direction of the proposed extension of jetties (Figure 6(c)), waves approaching from the west-north-west direction dissipate more wave energy before reaching the proposed jetties. As such, waves approaching from the west and west-south-west directions generate more H_s (0.02–0.9 m) at the proposed jetties as compared with the H_s (0.02–0.5 m) generated due to the waves approaching from the west-north-west direction. The result also indicates that under the existing condition, during the non-monsoon season, there will be downtime of about 30 days wherein the H_s near the proposed extension of jetties will be more than 0.3 m. As such, it is required to extend the existing bund so that the necessary tranquil condition is achieved. Studies were carried out by varying the length (55 m, 70 m) of existing bunds so that the optimised length of extension of the bund is achieved wherein the wave heights at the proposed extension of jetties remain under the permissible limit (0.3 m). As the waves approaching from 247.5° and 270° N generate the maximum H_s along the proposed extension of jetties as compared with the remaining predominant wave direction, the suitability of extension of the bund was primarily checked for the waves approaching from 247.5 and 270° N. Once the wave conditions for the proposed extension of the bund are found within the permissible limit, its suitability is also checked for the remaining predominant wave direction.

As such, to avoid downtime of the proposed extension of jetties, it is essential to extend the existing bund by 70 m. Thus, the assessment of reliable wave conditions plays an important role for the planning of construction activities for the development of the sea link bridge. A similar methodology can also be adopted to obtain reliable spectral characteristics of the wave atlas (i.e. H_s, T_p, T_m01, θ_m, γ, directional distribution of wave energy, etc.) wherein depending on the study area, the optimised regional model domain can be selected and forced with a spatio-temporally varying reanalysed 2D directional spectrum to save computational cost as well as to achieve a reliable wave climate at finer grid resolution. This wave atlas can be used for optimising the port layouts, selection of optimum length/orientation of protective structures, downtime analysis for maritime structures, shoreline evolution modelling, morphological evolution modelling in coastal areas, shelf-scale sediment transport modelling, assessment of tidal or wave energy potentials, etc.

The present study investigates the importance of assessment of reliable wave conditions for the extension of jetties to facilitate the transportation of bridge components to build a sea link bridge so that zero downtime for operability at jetties is ascertained. The following broad conclusions are drawn from the study:

• (i) Application of a numerical model forced with high-resolution reanalysed wind/wave/tide data is one of the reliable ways to hindcast the continuous historical wave climate. The multi-scaled nested model used for deriving the long-term historical wave climate along the Indian coast indicates that if the model is forced with only reanalysed wind, the entire IO needs to be modelled to simulate the appropriate seas-swell regime on the Indian coast. Thereby, a huge computational cost/time is required. However, the application of a coupled hydrodynamic wave model forced with spatio-temporally varying reanalysed 2D wave spectra/wind (obtained from ERA-5) and TPXO tide for a relatively smaller domain (region of the Mahim Bay, Mumbai) provides a reliable simulation of wave climate in the nearshore area and thereby it minimises the computational cost/time.

• (ii) The reanalysed 2D spectrum obtained from the ERA-5 database indicates that during non-monsoon season for most of the time, the offshore wave spectra are multipeaked, with seasonally varying γ (1.5–8.5) as well as there being variation in directional spreading of wave energy. In order to impose appropriate spectral density on the regional model, it is essential to incorporate the spatio-temporal 2D spectrum with varying γ along the offshore boundary of the regional model rather than imposing a specified spectrum with constant angular distribution function of wave energy. The comparison of measured and simulated H_s and T_m01 indicates that spatio-temporally varying reanalysed 2D spectrum imposed on the regional model provides better simulation of the wave conditions. The selection of appropriate discretisation of the directional spectrum is also essential for the proper calibration of the spectral wave model.

• (iii) The wave climate obtained from the regional model during non-monsoon season reveals that for most of the time in the nearshore areas (6 m depth) of the Mahim Bay, there is dominance of wind seas as compared with swell waves. The trend of H_s indicates that it gradually increases from the month of January to May and during post-monsoon seasons H_s gradually decreases except when there is an occurrence of extreme/cyclonic events. The maximum spectral density during pre- and post-monsoon seasons is mostly concentrated within the range of 3.8 and 5.4 s. As waves approach the nearshore areas, the spectral density decays by about 20%–30%, the mean or peak frequencies are increased by about 10%–20%, and there is also a reduction in the directional spreading of wave energy. Due to the dominance of short-period wind seas, during non-monsoon season, the direction of propagation of waves varies over a wider range (about 185°–340° N) and also there is the directional spreading of wave energy over a range of 30°–50°. Although the nearshore area (of the Mahim Bay) is situated in a macro-tide-dominated region, due to the presence of weak tidal current on the open coast, the impact of following/adverse current in modulating the wave condition is feeble.

• (iv) The analysis of trend of mean H_s and T_m01 for 22 years indicates that there is no increase in the trend of wave conditions due to the change in the climatological conditions. Therefore, the analysis of 22 years of data will provide sufficient information about the predominant wave climate for the extension of jetties wherein there is no necessity to extrapolate the wave climate to incorporate the impact of change in the climatological conditions.

• (v) The H_s of 1.0 m with T_m01 = 4 s, T_p = 5 s, and mean wave direction (θ_m) of 247.5°, 270° and 292.5° N were found to be the predominant wave climate (at 6 m depth) for the assessment of the wave tranquillity near the proposed extension of jetties.

• (vi) Wave agitation model studies were carried out by imposing the 2D spectrum (obtained from TOMAWAC) on the liquid boundary of the ARTEMIS model so that the impact of spatially varying γ and frequency spreading/directional spreading of wave energy are incorporated. Studies indicate that under the existing condition, the H_s along the proposed extension of jetties will be more for waves approaching from the west and west-south-west directions as compared with waves approaching from the west-north-west direction and the H_s will vary between 0.02 and 0.9 m for the predominant wave climate. Thereby, under the existing condition, during the non-monsoon season, downtime will be about 30 days wherein the H_s near the proposed extension of jetties will be more than 0.3 m. As such, to achieve the desired tranquil condition at the proposed jetties, it is essential to extend the existing bund.

• (vii) Studies carried out on extending the existing bund by 55 and 70 m indicate that the H_s at the proposed extension of jetties remains under the permissible limit (0.3 m) if the existing bund is extended by 70 m. As such, the application of spatio-temporally varying 2D wave spectra (with varying γ) as a forcing function to the regional model is essential in the determination of reliably accurate wave conditions for the proposed extension of jetties and thereby with an optimal extension of guide bund, zero downtime at the berth is ascertained.

The authors are thankful to Dr R. S. Kankara Director, Central Water and Power Research Station, Pune (India) for his continuous encouragement for carrying out the research work. The source code of the Telemac software suite was downloaded freely from the website of Open Telemac Mascaret. ERA-5 wind and wave data were also freely downloaded from the website of ECMWF organisation. The tidal database of the TPXO model was obtained from the website of Oregon State University, USA.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.