1 Introduction

Due to moist and warm air of equator, tropical cyclones are formed only over warm waters of oceans nearby the equator. Cyclones are necessary for the Earth’s atmosphere in spite of their demolishing effects, as they transfer energy and heat between the equator and the cooler areas nearby the poles, and cause rain to dry regions. The development of a tropical cyclone happens along the Inter-Tropical Convergence Zone (ITCZ), or monsoon trough, in the South-West Indian Ocean (Rhome and Raman 2006). Cyclones are related with steep pressure gradients and subsequently cause robust winds and storm surges.

Many countries in the North Indian Ocean are endangered by storm surge floods which are associated with intense tropical cyclones. Although the frequency of storm surges is much less in Arabian Sea compared to Bengal Bay. Tropical cyclones often happen in the Arabian Sea in the monsoons’ transition periods between October and November and between May and June. Cyclones are generated in the north of Indian Ocean from 5° to 20° north and 55° to 90° east (Webster et al. 2005). Furthermore, Gonu is the most robust tropical cyclone, which is recorded in the Arabian Sea (Fritz et al. 2010). It should be noted that atmospheric aspects and oceanographic responses to Gonu have been studied by different researchers (Allahdadi et al. 2017, 2018; Dibajnia et al. 2010; Fritz et al. 2010; Golshani and Taebi 2008; Jayakrishnan and Babu 2013; Krishna and Rao 2009; Mashhadi et al. 2013; Wang et al. 2012). With a maximum wind speed of 240 km/h on 5 June, Gonu was the strongest tropical cyclone that had ever occurred in the Arabian Sea. The cyclone moved northwest and its first landfall was over the eastern tip of Oman on 6 June. Then, it continued its path and its final landfall was on the Makran coast of Iran about 150 km east of the Persian Gulf (Al Najar and Salvekar 2010). The storm track is shown in Fig. 1a. Based on the reports from World Meteorological Organization (WMO), maximum significant wave height during Gonu Storm was achieved 4.2 m along Iranian Makran coasts (Fig. 1b), 8 m in the Gulf of Oman, in excess of 11 m in the Arabian Sea (WMO 2009); Also, Oman Meteorological Office reports indicated that maximum significant wave height in the Arabian Sea has been observed between 6 m and 12 m (Sarker 2018). Gonu cyclone caused 50 deaths in Oman and 28 along the Iranian Makran coasts. The economic damages were about $4 billion in Oman and more than $200 million in Iran (Fritz et al. 2010).

Fig. 1
figure 1

a Storm track of Gonu cyclone from 2 to 7 June; b Iranian Makran coasts and locations of Chabahar and Nikshahr weather Stations; c The location of measuring instruments in the Gulf of Oman (Chabahar Bay) during the Gonu cyclone

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For simulating the process of storm surge disaster in the numerical model, we should not only consider the pressure and the path of tropical cyclone, surface wind stress, the coupling effects between storm surge and astronomical tide, and coastal topographic feature, but also consider various factors, including the interface and interaction process of ocean and atmosphere, the coupling of wave and water level setup, which were illustrated by Feng et al. (2011), Kim et al. (2008), Wolf (2009), Mellor (2008), Roland et al. (2009), Lee et al. (2013) and Bertin et al. (2012). Recently, Fritz et al. (2010) numerically simulated storm surges according to ADCIRC and the analytical cyclone model to evaluate the hydrodynamic response to Super Cyclone Gonu in the Gulf of Oman. Mashhadi et al. (2013) simulated cyclone waves in the Arabian Sea using the SWAN model during Gonu tropical cyclone in 2007. For calculating the process of storm surge in the Arabian Sea while focusing on the cyclone of Chapala, Sarker (2018) employed a surge–wave coupled model.

Although wind is the major force driving the surge, waves and tide are also critical factors that affect the momentum and mass transport during the storm surge (He et al. 2020). Regarding the contribution of wave-induced setup on storm surge height, neglecting wave–current interactions can lead to inaccurate predictions of the storm surges (Li et al. 2020; Rautenbach et al. 2020). The effect of wave–current interaction on Gonu storm surge is not considered in the previous studies. The purpose of this study is development of the two-way coupling of wave and hydrodynamic models. In this study, a storm surge–wave–tide coupled 3D model was developed to evaluate the hydrodynamic response in the Gulf of Oman to Gonu tropical cyclone; this model considers multi-scale processes and multi-coupling factors. Through comparison of simulated water level and ocean wave characteristic with field observed during Gonu cyclone, the model was validated. For assessing the impact of the storm surge on Iranian Makran coastal region in the Gulf of Oman, wave conditions and the water level changes induced by storm surge are provided and discussed. Also, the effects of the interaction of the significant factors (wind, atmospheric pressure, and wave) on a storm surge in the Iranian Makran coasts were investigated.

2 Generation of Wind and Pressure Fields

The parametric tropical cyclone models and globally gridded data set are the two main sources to acquire wind and pressure fields to estimate wind waves and storm surge. Due to their accessibility and reliability, globally gridded data set such as NCEP Climate Forecast System Reanalysis (CFSR v.1) are extensively utilized (Shao et al. 2018). However, notably, the global wind datasets based on satellite remote sensing assimilation including NCEP CFSR v.1 and ECMWF Reanalysis Interim (ERA-I) are often lower than those of actual values close to the cyclone center and perform poorly in reproducing the wind speed around the tropical cyclone center (i.e., inner core) (Signell et al. 2005; Cavaleri and Sclavo 2006; Pan et al. 2016). Alternatively, various parametric cyclone models are widely employed worldwide for the wave and storm surge associated studies (Wijnands et al. 2016; Olfateh et al. 2017). However, these parametric wind models have better reproducibility in wind fields (Pan et al. 2016).

To produce the cyclonic pressure and wind fields of Cyclone Gonu, the MIKE21 Cyclone Wind Generation Tool of DHI (DHI 2017a) was employed. The tool permits users to calculate pressure and wind data of tropical cyclone. Some parametric models of cyclone are involved in the DHI tool such as the Rankine vortex model, Holland single vortex model (1981), Holland double vortex model (1980), and Young and Sobey model (1981) (DHI 2017a). The best track of the Gonu cyclone was achieved from the Joint Typhoon Warning Center (JTWC), USA (JTWC 2019). The data of cyclone archived by JTWC involve 6-hourly data. The JTWC archived cyclone data provided all six input parameters needed by the Young and Sobey model (i.e., track, time, maximum wind speed, central pressure, radius of maximum wind speed, and neutral pressure), and this was applied to produce the pressure fields and cyclonic wind. Other models need further variables (e.g., Rankine parameter X and Holland shape parameter B) that should be computed by empirical relations. Since this addition increases uncertainty in the produced pressure and wind fields, these models are less favorable.

The simulations was conducted over a period of 30 days from May 15th to June 15th, 2007.The meteorological data required in the hydrodynamics and wave models include wind fields in both x and y directions at the height of 10 m \((u_{{10}} .v_{{10}} )\) and pressure field. Before and after the Gonu cyclone, the NCEP Climate Forecast System Reanalysis (CFSR) v.1 data (Saha et al. 2010) were used for the wind and pressure fields that had a spatial resolution of \(1/3^\circ \left( {\sim 20\;\min } \right)\) and a temporal resolution 6-hourly products from May 15th to June 2th, 2007, and from June 8th to June 15th, 2007. It should be noted that because the NCEP meteorological data were not sufficiently accurate during the time period of the Gonu cyclone (June 3th to June 7th), the wind field and pressure of the Gonu cyclone using the Cyclone Wind Generation (CWG) tool of Mike DHI were generated from the storm parameters. This tool computes the pressure and wind data of the tropical cyclone. The spatial and temporal resolution of the produced data for the period of the storm was \(1/15^\circ \left( {\sim 4\;\min } \right)~\) and 1 h, respectively. The wind model products are summarized in Table 1.

Table 1 Review of wind products
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3 Numerical Modeling

To evaluate storm surges, two conventional physics-based numerical models are as following: a coupled model of surge, wave, and tide and a decoupled model of storm surge. Coupled models have been considered in the last three decades, particularly focusing on the interaction of wave, surge, and tide. The numerical models utilized in present study are portion of MIKE by DHI suite of models. The high number of simulations is needed for optimization of CPU time through running the models independently, though it is possible to run the models in a coupled mode. Although there are a significant feedbacks among the models in the near-shore regions and the sea state that influences the wind stress (such as Meza-Padilla et al. 2015; Mastenbroek et al. 1993; Moon 2005; Brown and Wolf 2009; Bertin et al. 2012; Olabarrieta et al. 2012), it was regarded that the errors that are induced would be lower compared to errors derived from the inaccuracies of near-shore bathymetry. The Hydrodynamic and Spectral Wave modules are simultaneously run as they feed back into one another at the selected master time step. Hence, since they share some similar or same inputs, they are considered here together. The storm surge–wave–tide two-way coupled three-dimensional model used in the Arabian Sea and Gulf of Oman is based on MIKE 3 HD FM and MIKE 21 SW FM, mainly for modeling of wave–current interaction and time-varying water depth. Figure 2 depicts the inputs to the Hydrodynamic and Spectral Wave modules, their outputs, and interactions.

Fig. 2
figure 2

Inputs to, interactions between, and outputs from the Spectral Wave and Hydrodynamic modules of Mike DHI when run simultaneously

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4 Bathymetry and Meshes

In the northern Indian Ocean, the computational domain covers the Arabian Sea. The simulation region includes the Arabian Sea and Gulf of Oman for the Gonu tropical cyclone-induced storm surge model. Figure 3 displays the simulation area in the computational domain. The computational grid extends from 14° N to 32° N and 47° E to 75° E. Bathymetry data were extracted from ETOPO 2 (Smith and Sandwell 1997) for computational grid regions (Arabian Sea) with 4-min spatial resolution, which is applied in many studies in the world because of its reliability. After smoothing with 270 × 420 nodes, these data were introduced into the model. In the mesh generation, a strategy of large domain-localized mesh resolution is utilized, defining large computational domain covering the major landed area of cyclone and locally refining the concerned regions with small triangular meshes. A flexible horizontal triangular mesh was constructed for each basin with 10 km grid resolution in the open boundary and offshore to finer elements close to the shoreline (\(\sim 250\;{\text{m}}\)) (Fig. 3). The horizontal triangular mesh has 58,727 elements and 36,822 nodes. The flexible mesh permits us to pinpoint the highest resolution to the region of interest, leaving the remainder of the model with a coarse grid so that it enhances the performance of computation. Vertical mesh is characterized as sigma mesh type and 10 equidistance vertical layers.

Fig. 3
figure 3

Model extent, mesh and bathymetry for in the computational domain

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5 Hydrodynamic Model

To simulate the water surface elevation produced by Gonu tropical cyclone, the hydrodynamic model MIKE 3 HD FM developed by DHI was applied which works on an unstructured mesh and cell-centered Finite-Volume 3D Model. The governing equations and MIKE 3 HD FM details can be seen in the MIKE 3 FM scientific documentation given by DHI (2017b). The period of simulation in the HD model varies based on the studied event; however, the time step is according to a multi-sequence integration step, indicated with a maximum 1800 s value and a minimum value of 0.01 s, and every time step is described on the basis of the Courant–Friedrichs–Levy (CFL) condition. The time integration and space discretization are of low-order fast algorithm, assuming a critical 0.80 CFL number. The tidal potential is a force, produced by changes in gravity due to the relative motion of the moon, earth, and sun. The forcing acts all over the computational domain. The tidal potential is described by the constituents that should be involved; every constituent is determined by some parameters. The default case involves 11 constituents which are \({\text{M}}_{2}\), \({\text{O}}_{1}\), \({\text{S}}_{2}\), \({\text{K}}_{2}\), \({\text{N}}_{2}\), \({\text{K}}_{1}\)., \({\text{P}}_{1}\), \({\text{Q}}_{1}\), \({\text{M}}_{{\text{m}}}\), \({\text{M}}_{{\text{f}}}\), and \({\text{S}}_{{{\text{sa}}}} .\) The last model setup assumed a baroclinic density model with a varying Coriolis force based on the domain, which has a constant \(0.28\;{\text{m}}^{2} /{\text{s}}\) eddy viscosity under the Smagorinsky formulation, and a constant \(32\;{\text{m}}^{{1/3}} /{\text{s}}\) bed resistance, employing the Manning number. Drag coefficient parameterization in the hydrodynamic model is associated with the model presented by (Wu 1980, 1994). As the drag coefficient increases with the wind speed to reach a maximum value at about \(30\;{\text{m}}/{\text{s}}\) (Moon et al. 2008; Takagaki et al. 2012), the parameterization was done on the basis of a fixed value of \(0.0012\) wind friction coefficient for speeds of wind under \(7.5\;{\text{m}}/{\text{s}}\) and a fixed \(0.0024\) value for higher than \(27.5\;{\text{m}}/{\text{s}}\) speeds, with a linear wind friction variation between them. Lateral boundary conditions are assumed to be zero for normal flow to the solid boundary, and along the open boundary is located at a latitude of 14° N. In the hydrodynamic model, the values of water level for open boundaries along the 14° N of the HYCOM global model with \(1/12.5^\circ (\sim 4.8\;{\text{min}})\) are obtained, then it is interpolated with \(1/15^\circ \left( {\sim 4\;{\text{min}}} \right)\) spatial resolution and a temporal resolution of 30 min that was provided for the boundary of the model. Moreover, water level tidal values for the open boundary consist of eight harmonic parameters of the main tidal components (\({\text{O}}_{1}\), \({\text{S}}_{2}\), \({\text{K}}_{2}\), \({\text{N}}_{2}\), \({\text{K}}_{1}\), \({\text{P}}_{1}\), \({\text{Q}}_{1}\), and \({\text{M}}_{2}\)), which was simulated applying the OSU Tidal Prediction Software (OTPS) (Egbert and Erofeeva 2002) with 30-min temporal resolution and \(1/15^\circ \left( {\sim 4\;{\text{min}}} \right)\) spatial resolution. The values which were interpolated from the OTPS and HYCOM models were added together and introduced to the hydrodynamic model. In implementing a baroclinic condition, salinity values and sea temperature are required in addition to velocities and water level of the flow at the boundaries. Salinity and temperature parameters were derived from the global model of HYCOM for a 1-month period and then interpolated into 10 layers (vertical water column) with an accuracy of 50 m. The initial state of condition is the elevation of surface; it was derived from HYCOM + NCODA Global \(1/12.5^\circ (\sim 4.8\;{\text{min}})\) Reanalysis model with 86,400-s temporal resolution. The cyclone effects can be taken into account until the stable field of flow is achieved. After 50-m spatial interpolation, initial temperature and salinity conditions were derived from the HYCOM model and interpolated to the model. Through activating the temperature/salinity module in the baroclinic implementation of the model, heat exchange with the atmosphere is also included. The heat in water can be transferred to the atmosphere through heat exchanges at the surface. If heat exchange is included, the atmospheric condition must be specified. Atmospheric condition comprises of near-surface air temperature at 2-m height above the sea surface, relative humidity and clearness coefficient that were introduced in the model by extracting from ECMWF Reanalysis Interim (ERA-I) data.

6 Wave Model

For simulating wave growth, decay, and waves transformation produced by wind as a result of each synthetic event, the third-generation spectral wave model MIKE 21 SW FM was employed (Sørensen et al. 2005). The reader is referred to DHI (2017c) for more information about the model. In the model, the spectral and non-stationary time formulation was employed, with a logarithmic spectral discretization with 0.04 Hz minimum frequency, a frequency factor of 1.1, 18 frequencies, and a directional discretization for 360° which was divided into 18 directions. With a minimum 0.01 s and a maximum 3600 s, the time step is on the basis of a multi-sequence integration step. The transfer of energy consists of the wave-breaking factor and interactions of quadruplet wave is an alpha value of 1.0 and a constant 0.80 gamma value. With a constant roughness value of 0.04 m, the friction of the bottom was according to the Nikuradse roughness. Whitecapping is controlled through the tunable coefficient constant values with a \(C_{{{\text{ds}}}}\) value of \(1.36 \times 10^{{ - 5}}\) and a \(\delta\) set to 0.6. The initial condition utilizes the JONSWAP fetch growth expression with shape parameters \(\gamma = 3.3,~\sigma _{{\text{a}}} = 0.07\;{\text{and}}\;\sigma _{{\text{b}}} = 0.09\). The boundaries of offshore are assumed opened (i.e., open boundary is at a latitude of \(14^\circ \;{\text{N}})\). Thus, the ECMWF Reanalysis Interim (ERA-I) data were used to extract the wave parameters (wave direction, significant wave height, and period) on the open boundary with a 6-h temporal resolution and spatial resolution of \(1.5^\circ \left( {90^{\prime}} \right)\), and they were proposed as boundary conditions at the 19 points along the open boundary.

7 Numerical Experiment Design

Four numerical experiments to quantify the contribution of wind, atmospheric pressure, and wave to a storm surge are designed in this study based on a tide–surge–wave two-way coupled numerical model (Table 2). Experiment 0 is the recommended experiment explained above. Experiment 1 ignored wind to analyze the effects of wind on the total surge. Atmospheric pressure was excluded in Experiment 2 to investigate the influences of atmospheric pressure on total surge. Experiment 3 removed wave radiation stress to examine the wave’s contribution on the total surge. In experiment 4, only the astronomical tide is used to calculate the total surge during Gonu super cyclone.

Table 2 Descriptions of numerical experiments
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8 Results and Discussion

8.1 Model Results and Discussions

In 2006, the first stage of the project was started on monitoring and modeling of the coasts of Iran (Chabahar Bay case study) by the I. R. IRAN Ports and Maritime Organization (PMO). This phase of the project was involved in field measurements for 1 year (September 2006 to August 2007) in the Chabahar Bay. These measurements involved current and wave measurements at six stations, involving measurements of water level at three stations, and measurements of wind at one station. Figure 1c depicts the stations of measurement in the Bay of Chabahar in the Gonu cyclone. These data are the only data available in the Gulf of Oman in the time of Gonu cyclone. To validate and calibrate modeling, AW1, AW2, AW3, TG1, TG2, and TG3 instruments data on the coast of Makran were used (Table 3). For 31 days from May 15 to June 15, 2007, the simulations of coupled 3D model were done.

Table 3 The location of the AW1, AW2, AW3 (Nortek AWAC), TG1, TG2 and TG3 (tide gauge) instruments used for model validation
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The most practical and appropriate tunable parameter in third-generation wave models is related to the wind input source term (Cavaleri et al. 2007; Lee 2015). But, in the third-generation wind wave model, the whitecapping parameterization also acts a like a tuning knob in the action balance equation. Whitecapping coefficients are one of the important calibration parameters in the investigations. Also, acceptable outcomes can be achieved via whitecapping tuning coefficients. The coefficients for whitecapping dissipation were differed around their default setting, and then model was ran. The modeled significant wave height data for each setting (\({\text{various}}\;C_{{{\text{ds}}}}\)) were compared with the observed data at the AW1 station and then statistical error was estimated. Calibration was done based on minimizing Root Mean Square Error (RMSE) and a higher level of correlation (\(R^{2}\)) in \(H_{{\text{S}}}\) simulation which is more important than \(T_{{\text{P}}}\). In this study, according to Fig. 4, \(C_{{{\text{ds}}}} = 1.36 \times 10^{{ - 5}}\) was selected in simulation.

Fig. 4
figure 4

RMSE and R2 criteria variation with respect to \(C_{{{\text{ds}}}}\) coefficient alteration in calibration

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In the calibration method of hydrodynamic model, Manning’s coefficient is adjusted to reach the best match between observed measurements and modeled water level in TG2 station. Based on the range of variation for both RMSE and \(R^{2}\) values in the Fig. 5, a Manning’s coefficient of about 0.031 was selected as an optimum.

Fig. 5
figure 5

RMSE and R2 criteria variation with respect to Manning’s coefficient alteration in calibration

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For validation of the modeling results, the wave observation data were utilized which were recorded by the AW1, AW2, and AW3 devices located in the Bay of Chabahar from May 15 to June 15. Figures 6, 7, 8 demonstrate the comparisons results of the wave characteristics achieved from simulation and observational data recorded on these devices. Figure 6 shows that the significant wave height (\(H_{{\text{S}}}\)) of the model is very accurate. Figure 7 depicts that the simulation of mean wave direction (MWD) results are accurate in AW2 and AW3 instruments. Nevertheless, in the case of the AW1 instrument, it is overestimated. According to Fig. 8, the simulated peak period (\(T_{{\text{P}}}\)) is underestimated compared to those of all instruments data for all periods, excluding Gonu storm period. This might be because of the absence of long-period swell waves in the modeling. Besides, because of its geographical location, the southern coast of Iran is close to the open seas and is influenced by the southern swell waves (Kumar et al. 2011). The study region is related to the northern hemisphere, and the swell waves from Antarctica are not in this modeling. Additionally, the waves height traveling from the south is reduced over time; however, their period does not change (Hassannezhad et al. 2011). Thus, an error source could be the boundaries of modeling area; the boundary conditions gained from the ECMWF data. Another reason for mismatches could be inadequate accuracy of temporal and spatial wind data. Statistical indices are discussed to estimate the overall performance of the model in the following.

Fig. 6
figure 6

Time series of the observed and simulated significant wave height at the a AW1, b AW2, and c AW3 stations

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Fig. 7
figure 7

Time series of the observed and simulated mean wave direction at the a AW1, b AW2, and c AW3 stations

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Fig. 8
figure 8

Time series of the observed and simulated peak period at the a AW1, b AW2, and c AW3 stations

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The difference in the water level versus the tide indicates the effect of the wind field in the variation of the water level. In this paper, the tidal analysis and prediction module of Mike 21 has been used to determine the magnitude of storm-induced fluctuations. By removing tidal components from the water level, storm surge is obtained.

The model is validated against measured water level data from three tidal elevation stations in fair condition, and it can simulate storm tide quite accurate in the Gulf of Oman. Time series of water level in the cyclone period of three tidal elevation stations around the Bay of Chabahar are depicted in Fig. 9. The measurements are shown by the solid dots; the red solid lines show the levels of water computed with all the dissipation and forcing agents, including TG1 and TG3 station starting from 00 o’clock on May 15, 2007 and TG2 station starting from 18 o’clock on June 3, 2007. Due to the technical problems occurred during the measurement period, the time series of measurements have discontinuities and sometimes do not have the same start and end time. Figure 10 represents residual component time series at TG1, TG2 and TG3 stations. It is seen that the computed water levels agree reasonably well with the observational measurements at each tidal elevation station. The model underestimated the water level slightly at peaks of the TG2 station for rainfalls caused by Gonu cyclone that raised the Wadi flooding and overland flow discharge. The mountainous and arid terrain near the coasts of Oman and Iran pose an additional hazard to coastal regions along the Gulf of Oman. Gonu dumped substantial amounts of rain in southern Iran with locally heavy amounts. From June 6th to 9th 2007, in nearest meteorological stations to Iranian Makran coasts, including Chabahar and Nikshahr synoptic weather stations, the cumulative precipitations were reported about 108.6 and 143.8 mm, respectively. Heavy rain falling on the steep mountains sent torrents of fast-moving floodwater down to the coastal areas (Fig. 1c).

Fig. 9
figure 9

Time series of the observed and simulated water level at the a TG1, b TG2 and c TG3 stations

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Fig. 10
figure 10

Time series of the observed and simulated residual at the a TG1, b TG2 and c TG3 stations

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The maximum difference is about 0.25 m between the measured and computed levels of water at TG1 Station. These variations may be induced by the inaccurate wind stresses. From all stations, the mean relative error of the extreme values is below 10%. The coupled 3D model of storm surge–tide–wave could well simulate the influences of surface waves and local wind forcing on the storm surge dynamics.

8.2 Skill Assessment Indices

To evaluation of the overall model performance, statistical indices are considered. The comparison of the model prediction with actual observations is used to verify the simulation accuracy. To achieve more concise verifications, the statistical skill indices can be utilized. Willmott (1982) used normalized bias, bias parameter, scatter index (SI), root mean square error (RMSE), the standard deviation, correlation coefficient (R), and index of agreement (d) to assess the consistency between two time-series datasets (e.g., in situ observations and model prediction). These statistical parameters were calculated as follows with the relations (1)–(6), and the results are shown in Tables 4 and 5:

$${\text{Bias}} = \frac{1}{k} \times \mathop \sum \limits_{{i = 1}}^{k} \left( {m\left( i \right) - {\text{obs}}\left( i \right)} \right),$$
(1)
$${\text{Normalized}}\;{\text{bias}} = \frac{1}{k} \times \mathop \sum \limits_{{i = 1}}^{k} \left| {\frac{{m\left( i \right) - {\text{obs}}\left( i \right)}}{{{\text{obs}}\left( i \right)}}} \right|,$$
(2)
$${\text{RMSE}} = \sqrt {\frac{1}{k} \times \mathop \sum \limits_{{i = 1}}^{k} \left( {m\left( i \right) - {\text{obs}}\left( i \right)} \right)^{2} } ,$$
(3)
$${\text{St}}\_{\text{Dev}} = \sqrt {\frac{1}{k} \times \mathop \sum \limits_{{i = 1}}^{k} \left( {m\left( i \right) - {\text{obs}}\left( i \right) - {\text{Bias}}} \right)^{2} } ,$$
(4)
$${\text{Scatter}}\;{\text{Index}} = \frac{{{\text{Standard}}\;{\text{deviation}}\;{\text{of}}\;{\text{error}}}}{{{\text{Mean}}\;{\text{observed}}\;{\text{significant}}\;{\text{wave}}\;{\text{height}}\left( {{\text{or}}\;{\text{other}}\;{\text{parameters}}} \right)}},$$
(5)
$$d = 1 - \frac{{\mathop \sum \nolimits_{{i = 1}}^{k} \left( {m\left( i \right) - {\text{obs}}\left( i \right)} \right)^{2} }}{{\mathop \sum \nolimits_{{i = 1}}^{k} \left( {\left| {m\left( i \right) - \overline{{{\text{obs}}}} } \right| + \left| {{\text{obs}}\left( i \right) - \overline{{{\text{obs}}}} } \right|} \right)^{2} }}~0 \le d \le 1.$$
(6)
Table 4 Statistics describing the performance of wave parameters simulations for all observation stations
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Table 5 Statistics describing the performance of water level simulations for all observation stations
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In these relationships, \({\text{obs}}\left( i \right)\) represents the observed value at time i, \(\overline{{{\text{obs}}}}\) is the average observation values, \({\text{m}}\left( i \right)\) represents the respected value of the model prediction, k is the sample size, and ‖ is absolute values and shows the divergence of the forecasts as a proportion of observations.

Several statistical indices are necessary to prevent pitfalls related to use of a single parameter. For instance, the mean of two time-series datasets is compared by the bias, hence describing the systematic error between two time series and excluding the distribution data around the value of the mean. Root mean square error clarifies the scatter of model results around the observations; however, it is not bounded and it is not simple to make decision when it is small. Definitely, the calculated statistical skill indices in Tables 4 and 5 verify the results of Figs. 6, 7, 8, 9, 10.

Bias measures the average tendency of the simulated constituent values to be larger or smaller than the measured data.

Bias of significant wave height at three stations AW1, AW2, and AW3 and at three periods of pre-storm, during Gonu storm, and post-storm is close to zero. In the peak period, the bias values in the Gonu storm period are close to zero, but for the main wave direction, the bias values in each of the periods are relatively large and are not acceptable. Residual variance is difference between the measured and simulated value, often estimated by the residual mean square or root mean square error (RMSE). RMSE value of 0 indicates a perfect fit. RMSE values for significant wave height at three stations and in three periods are close to zero. In the peak period, the values of this statistical index are somewhat acceptable, but they are not quite suitable for the main wave direction. Standard deviation can be difficult to interpret as a single number on its own. Basically, a small standard deviation means that the values in a statistical data set are close to the mean of the data set, on average, and a large standard deviation means that the values in the data set are father away from the mean, on average. Standard deviation values for significant wave height at three stations in three periods are close to zero. In the peak period, the values of this statistical index are somewhat acceptable, but for the main wave direction, the values of this skill assessment index are not suitable at all situations. The average scatter index for significant wave height at three stations over three time periods is about 0.18 (18%). This value is about 0.16 (16%) for the peak period and about 0.09 (9%) for the main wave direction, which are almost acceptable results in three situations. Willmott (1981) proposed an index of agreement (d) as a standardized measure of the degree of model prediction error which varies between 0 and 1. The agreement value of 1 indicates a perfect match, and 0 indicates no agreement at all. The index of agreement can detect additive and proportional differences in the observed and simulated means and variances; however, d is overly sensitive to extreme values due to the squared differences. Application of the index of agreement shows that the relatively high values of d may be obtained even for a poor model fit. Evaluation of this statistical index in Table 4 indicates that the values of the index of agreement in all cases are higher than 0.8, but the highest values of this index are calculated for significant wave height (Table 4).

Assessment of the statistical indicators calculated in Table 5 shows that at different stations and periods, the values of these skill assessment indies are closer to acceptable values.

8.3 Contribution Wind, Atmospheric Pressure and Wave to the Storm Surge

In the current study, the contribution of each of the winds, atmosphere pressure, and wave factors to storm surge in the Iranian Makran coasts was assessed using numerical experiment designs. The wind-induced surge was calculated by comparing the water levels between experiments 0 and 1 (Table 2). Figure 11 indicates the water levels of experiments 0 (black solid line) and 1 (red solid line). Also, their differences between experiments 0 and 1 (wind-induced surge, blue solid line) at TG1 station are represented in Fig. 12. The peak value of the wind-induced surges at the TG1 station were 1.82 m and 77.9% of the total storm surge. The total storm surge during Gonu storm was obtained by comparing the differences of water levels between the recommended experiment (experiment 0) and experiment tide (experiment 4) (Table 2).

Fig. 11
figure 11

Water levels of the experiment 0 (black dash line), experiment 1 (red solid line), experiment 2 (blue solid line), and experiment 3 (green solid line) at the TG1 station

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Fig. 12
figure 12

The differences experiments 0 and 1 (wind-induced surge, blue solid line), the differences experiment 0 and 2 (pressure-induced surge, red solid line) and differences experiment 0 and 3 (wave-induced surge, black dotted line) and a peak surge induced by winds (red square point), peak surge induced by air pressure (black circular point) and peak surge induced by ocean wave (green triangular point) at TG1 location

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The atmospheric pressure-induced surge was assessed by comparing the water levels of experiments 0 and 2. Figure 11 indicates the water levels in experiments 0 (black solid line) and 2 (blue dotted line). Also, their differences between experiments 0 and 2 (atmospheric pressure-induced surge, orange solid line) at TG1 station is represented in Fig. 12. The peak of the air pressure-induced surges at the TG1 station were 0.51 m and corresponded to about 22.3% of the total storm surge.

The wave-induced surge was evaluated by comparing the water levels of experiments 0 and experiment 3 (Table 2). Figure 11 again indicates the water level in experiment 0 (black solid line) and experiment 3 (red solid line). Also, the differences experiments 0 and 3 (wave-induced surge, black dotted line) at TG1 station in Fig. 12. The peak of the wave-induced surge at the TG1 station was about 0.24 m (10.3%).

A peak surge induced by winds occurred first (red square point in the Fig. 12), followed by peak wave-induced (green triangular point in the Fig. 12) and the air pressure-induced surges (black circular point in the Fig. 12).

8.4 Concluding Remarks

A three-dimensional storm surge–wave–tide two-way coupled model is employed in this study to investigate the hydrodynamic response in the Gulf of Oman to Gonu tropical cyclone. The model was well calibrated using field data of water level and significant wave height. Good-quality observed data are needed for model calibration and validation. Here, all good observational data were acquired from Iranian Port and Maritime Organization (PMO). Based on this validated model, the storm surge induced water level changes and wave conditions that are utilized to evaluate the impacts of the storm surge on the Makran coasts in North of Gulf of Oman. Research performed in this paper accomplished the development of coupling model interface, which permitted not only the wave–current interaction but also the non-linear relationship among tides, wind surges and wind-induced waves. By numerically examining the contributions of wind, air pressure, and wave to the storm surge, the physical drivers of the storm surge during the Gonu cyclone were considered.

The comparison of the wave simulation results with the observed data in all three stations shows that the significant wave height (\(H_{{\text{S}}}\)) was well matched. The mean wave direction (MWD) in stations AW2 and AW3 were in accordance with field data during the modeling period, but in station AW1, the modeling results have been overestimated. The simulated peak period (\(T_{{\text{P}}}\)) at all three stations is underestimated except during Gonu tropical cyclone period.

The simulated water level in all three stations TG1, TG2, and TG3 was in good agreement with the observed data. Since the Gonu storm reached the Iranian Makran coasts on June 6, after weakening and losing its energy, and due to the coincidence with the neap tide, the resulting storm tide has not been significant; therefore, in case of more severe tropical cyclone in future and its coincidence with spring tide, it can cause irreparable economic and environmental damage to this region.

Wind has a prevailing impact on a storm surge, while wave–current interaction has a minor contribution to the total surge. However, despite the low contribution of wave–current interaction, it cannot be ignored to achieve accurate forecasts. Wind, pressure, and wave contributed 77.9%, 22.3%, and 10.3%, respectively, to the surge of storm during Gonu tropical cyclone.

Finally, the coupling of hydrodynamic and wave models investigates the wave–current interaction in the storm surge. The conclusions from this study have implications toward the refinement of storm surge modeling.