1 Introduction

The east coast of India (ECI) is characterized by a highly productive estuarine environment, mangrove ecosystem and thickly populated coastal cities. Low-lying coastal areas along the ECI are highly susceptible and vulnerable to the impact of storm surges due to landfalling tropical cyclones that originate in the Bay of Bengal basin. The national weather agency has categorized both Phailin and Hudhud cyclones as very severe cyclonic storms (VSCS) that made landfall along the ECI near Gopalpur, Odisha and Visakhapatnam, Andhra Pradesh, respectively. These two VSCS resulted in a trail of destruction during landfall, thereby causing widespread damage to infrastructure, property and a few instances of loss of human life in the coastal regions of Odisha and northeastern Andhra Pradesh. Coastal flooding or inundation is a catastrophe having significant impact on any coastal zone, causing loss of life and property and widespread damage to coastal infrastructure. Timely warnings of storm surge and inundation forecasts during cyclones can potentially reduce the loss of life by issuing bulletins and alerts to stakeholders. Numerical modeling of coastal inundation during cyclonic events is a vital element in issuing warnings for emergency planning and evacuation measures. While several studies had focused on storm surge modeling along the ECI (Murty et al. 1986; Dube et al. 1997, 2000; Rao et al. 1994), very few studies have focused on modeling inundation scenarios during storm surge events. Rao et al. (2012) investigated coastal inundation scenarios along the ECI attributed due to tropical cyclones. Bhaskaran et al. (2014) performed a detailed analysis on the potential of coastal inundation at several coastal locations off Tamil Nadu. A study by Srinivasa et al. (2015) investigated the possible worst-case scenario by considering the combination of highest high water levels with storm surge that generates huge devastation during cyclones. Their study highlights the importance of the tidal phase and its relevance in controlling the extent of inland inundation during tropical cyclone landfall.

In a shallow-water environment, storm surge and wind waves can interact with each other by transferring energy from one system to the other. It is well known that wind waves can ultimately affect the coastal ocean circulation by enhancing the wind stress (Mastenbroek et al. 1993). Surface waves constitute sea surface roughness, and for a strong wind regime such as in tropical cyclones (> 30 m/s), they tend to decrease the surface drag (Hwang 2018) when other important processes start to play an important role (e.g. sea spray). The study by Hwang (2018) also indicates that both surface roughness and whitecaps due to wave breaking are driven by surface wind stress, and their quantification using the wind speed input depends on the drag coefficient formulation. A previous study by Rajesh et al. (2009) attempted to develop a parameterization of sea surface drag under varying sea states dependent on wave age. Their study (Rajesh et al. 2009) combined the Toba-3/2 power law and Froude number scaling to develop a new drag formulation found to perform well under conditions of both young and mature waves, verified against time series meteorological and oceanographic observations in the Indian Ocean region. Usually, these physical processes can influence one another in numerous ways: wind stress that is altered by including the wave effect (Donelan et al. 1993); wave radiation stress considered to be an additional forcing term in storm surge models (Xie et al. 2001; Mellor 2003; Xie et al. 2004). The concept of radiation stress was first introduced by Longuet-Higgins and Stewart (1962) to analyze the mechanism of wave setup and set-down in a nearshore environment.

In shallow waters, storm surges, tides and wind waves are the major components in an ocean forecasting system that determines the sea-state environment of that region. It is known that these processes are present in many cases and their interactions are very significant in context to individual dynamic processes (Longuet-Higgins and Stewart 1960; Prandle and Wolf 1978; Janssen 1992; Bao et al. 2000; Welsh et al. 2000). Wind waves play a significant role in determining the sea surface roughness that in turn affects the currents in coastal circulation models (Smith et al. 1992). Although short- and long-period waves are separated in a wave spectrum, they interact. Breaking waves in shallow water generate radiation stress gradients that in turn drive wave setup and currents affecting the vertical momentum mixing, bottom friction and the resultant water level elevations. According to the conservation of wave energy flux, the wave height and steepness have a tendency to increase as they propagate into shallow water environments. The shallow-water wave process is dominated by depth-induced breaking, wherein momentum flux is transferred to the water column, raising the water levels adjacent to the coast (Holthuijsen 2007). The dynamic characteristics of water level elevation and wave-induced setup and set-down along coastal areas, especially during an approaching storm, have cumulative effects that depend on the mutual interaction between waves, currents, surge and the astronomical tides. In fact, tidal propagation is also modified in the presence of a surge, leading to tide-surge interaction (Horsburgh and Wilson 2007), and the waves are modified by the presence of currents generated by the tide and surge. They contribute to resultant water level and mean circulation through wave-induced setup and currents, attributed to radiation stresses created by breaking waves in shallow water (Longuet-Higgins and Stewart 1962). The dynamic interaction between wave–current–surge–tide components and their combined energy balance is very strong during tropical cyclone activity. Hence, the nonlinear interaction mechanism and the influence of waves on coastal currents and storm surge has been an important area of research area in recent years. The key question lies in how these nonlinear processes can be properly accounted for and coupled in numerical models.

Application of coupled wave hydrodynamic models for the global ocean is a recent development that has been observed in a few studies such as the MAST III PROMISE Project (Ozer et al. 2000); the modeling of energetic events such as hurricane Katrina and Rita in 2005, and Gustav and Ike in 2008 in the Atlantic Ocean (Dietrich 2010); modeling of the Mediterranean Sea with focus on the Italian coast (Ferrarin et al. 2013); and a coupled modeling system for typhoon Maemi in Korean seas (Bung et al. 2013). Arora and Bhaskaran (2012) reported on the importance of bottom friction under combined wave-tide action and its relevance to bottom boundary layer characteristics (Arora et al. 2010). The role of wave-induced setup during an extreme event for the Kalpakkam coast (Nayak et al. 2012) and associated coastal vulnerability was studied by Nayak and Bhaskaran (2013). A few studies have emphasized the importance of coupling storm surges, waves and tides to enhance the accuracy of storm surge forecasts in coastal areas (Roland et al. 2009; Brown and Wolf 2009; Wolf 2009). During severe cyclones, in particular, radiation stress-induced wave setup is an important component which needs to be included in a storm surge model, as it can account for significant increase in water level along the coast (Brown et al. 2011). Araújo et al. (2011) studied the importance of including wave setup along the Viana do Castelo (Portugal) coast. A more recent study by Murty et al. (2014) and Murty et al. (2016) highlighted the significance of including wave setup in computing surge heights along the Odisha coast and for the West Bengal coast (Gayathri et al. 2016). However, their study does not investigate the effect of wave setup on inundation extent. Therefore, in the present study, the effect of wave setup on inundation extent is examined using a coupled wave and hydrodynamic model. The coupled model was used to study the characteristics of wave setup and inundation extent associated with two VSCS, the Phailin and Hudhud cyclones. The objective is to understand the effect of wave–current interaction on storm surges and inundation mutually transferred through wave radiation stress in storm surge computation.

2 Modeling System

The ADCIRC hydrodynamic model and coupled ADCIRC + SWAN model are used in the present study. Cyclonic wind fields were derived using the Holland dynamic wind model (Holland et al. 2010).

2.1 Wave Model: SWAN

The wind-wave model, Simulating Waves Nearshore (SWAN), was used to predict the evolution of surface gravity waves in time and geographic space. It computes the wave direction (\(\theta\)) and relative frequency (\(\sigma\)) based on the action balance equation (Booij et al. 1999), as given below:

$$\frac{ \partial }{\partial t}N + \frac{\partial }{\partial x}c_{x} N + \frac{\partial }{\partial y}c_{y} N + \frac{\partial }{\partial \sigma }c_{\sigma } N + \frac{\partial }{\partial \theta }c_{\theta } N = \frac{S}{\sigma }$$
(1)

where N represents the wave action density; S is the source term (combination of source and sinks) in terms of energy density; \(c_{\theta }\) is the propagation velocity in \(\theta\) space; \(c_{\sigma }\) is the propagation velocity in \(\sigma\) space; \(c_{x} \;{\text{and}}\; c_{y}\) are the respective propagation velocities along the x and y directions. The first, second and third terms on the left-hand side of Eq. (1) represent the rate of change of wave action in time and the propagation of wave action in geographic space, respectively. The current study used the unstructured SWAN model (version 40.85) coupled with a surge hydrodynamic model (ADCIRC). SWAN computes the wave action density spectrum at the unstructured mesh nodes by using a sweeping algorithm and updates the action density using information from the adjacent nodes. A detailed description of the SWAN model is available in Booij et al. (1999). In the current study the unstructured version of SWAN (UNSWAN) is used that implements an analog to the four direction Gausse–Seidel iteration method supplying unconditional stability (Zijlema, 2010).

2.2 Surge Hydrodynamic Model: ADCIRC

The advanced circulation (ADCIRC) model, developed at the University of North Carolina, is a multidimensional, depth-integrated, barotropic time-dependent long-wave finite-element hydrodynamic circulation model. The model can be run either as a two-dimensional depth-integrated (2DDI) or as a three-dimensional (3D) model. Both the vertically integrated (ADCIRC-2DDI) and the fully 3D (ADCIRC-3D) models solve the vertically integrated continuity equation for water surface elevation. A detailed description of ADCIRC is given in Luettich et al. (1992).

In the current study, the ADCIRC-2DDI version is used for the simulations. In the baroclinic version of ADCIRC-3D, the bottom boundary friction layer of surface Ekman wind will be better resolved as compared with the ADCIRC-2DDI (barotropic) version. However, as the radiation stress induced by breaking waves is confined to very shallow and nearshore waters, the use of ADCIRC-2DDI in the current study can be justified. However, for a detailed study on wave–current interaction in intermediate water depths, it would be better to study the simulation performed using the ADCIRC-3D model.

2.3 Coupling Methodology

The ADCIRC and SWAN models run on the same local unstructured mesh, and are coupled by passing mutual information from each other (Dietrich et al. 2011). Due to the sweeping method incorporated in the SWAN model, it usually larger time steps than the ADCIRC model (Dietrich et al. 2011).

ADCIRC runs first within the coupling interval and then passes the currents, water levels and wind velocity to SWAN during the prescribed coupled run. Complete details of the coupling procedure between the SWAN and ADCIRC models used in the present study are available in Dietrich et al. (2011). Using the wave radiation stress information from SWAN, water levels and the currents from ADCIRC are revised and thus mutually exchanged with SWAN to further update the radiation stress. The radiation stress gradient (\(\tau_{s}\)) computed by the SWAN model is expressed in the form:

$$\left. \begin{aligned} \tau_{{sx,{\text{wave}}}} = - \frac{{\partial S_{xx} }}{\partial x} - \frac{{\partial S_{xy} }}{\partial y} \hfill \\ \tau_{{sy,{\text{wave}}}} = - \frac{{\partial S_{xy} }}{\partial x} - \frac{{\partial S_{yy} }}{\partial y} \hfill \\ \end{aligned} \right\}$$
(2)

where \(S_{xx}\), \(S_{xy}\) and \(S_{yy}\) are the wave radiation stresses.

The two models march ahead in time by forcing each other by mutually exchanging the information. Figure 1 provides more details on the basic coupling algorithm between the SWAN and ADCIRC models.

Fig. 1
figure 1

Basic structure of the ADCIRC + SWAN coupling system

Full size image

2.4 Atmospheric Forcing

Surface wind stress and pressure gradients are the key driving forces for the computation of storm surges (Rao et al. 2012). Surface wind stress acts in parallel and pressure gradient in a direction normal to the sea surface, and their relative significance depends on the water depth. Parametric wind models and the winds from global numerical weather hindcast/forecast models are commonly used for storm surge and wave computations. Global/mesoscale models are not available at a high resolution in forecast mode to provide realistic cyclonic wind fields. Even robust meteorological nowcasts have preprocessing delays pertaining to huge input datasets (Fleming et al. 2008). In addition, the demand of computational resources is required to address the mesoscale and synoptic-scale convection processes in operational weather models, which limit their use as a regular operational forecast of hurricane winds (Smita and Rao 2018). On the contrary, parametric cyclonic wind models are easier to use, and they produce improved wind fields with desired spatial resolution for storm surge simulations (Houston et al. 1999). In this context, a dynamic wind model (Holland et al. 2010; hereafter referred to as H10) is used in the present study. H10 is the revised version of an earlier model (Holland 1980; hereafter referred to as H80). The H10 has several advantages over the earlier formulations, including (a) enhanced ability to contain both external and core wind observations; (b) less dependence on precise knowledge pertaining to radius of maximum wind, Rmax; (c) capability to handle bimodal wind profiles connected with secondary eye wall production; and (d) abridged matters with lack of resolution in some cyclones in the proximity of Rmax, which can lead to underrating the true maximum winds, or overrating external winds if the core is artificially improved (H80) (Holland et al. 2010).

Surface wind speed can be computed radially using the mathematical formulation:

$$v_{\text{s}} = \left[ {\frac{{100b_{\text{s}} \Delta p_{\text{s}} \left( {\frac{{r_{{v_{\text{m}} }} }}{r}} \right)^{{b_{s} }} }}{{\rho_{\text{s}} e^{{\left( {\frac{{r_{{v_{\text{m}} }} }}{r}} \right)^{{b_{\text{s}} }} }} }}} \right]^{x} {\text{or}}$$
(3)
$$v_{\text{s}} = v_{\text{m}} \left\{ {\left( {\frac{{r_{{v_{\text{m}} }} }}{r}} \right)^{{b_{\text{s}} }} e^{{\left[ {1 - \left( {\frac{{r_{{v_{\text{m}} }} }}{r}} \right)^{{b_{\text{s}} }} } \right]}} } \right\}^{x}$$
(4)

where \(v_{\text{s}}\) is the radial surface wind speed, \(v_{\text{m}}\) is the maximum wind speed, \(b_{\text{s}}\) is a scaling parameter, \(x\) is a shape parameter, \(r_{{v_{\text{m}} }}\) is the radius of maximum winds and \(r\) is the radial distance from the center.

\(b_{\text{s}}\), \(x\) and \(v_{\text{m}}\) values can be calculated as

$$b_{\text{s}} = - \;4.4\; \times \;10^{ - 5} \Delta p_{\text{s}}^{ 2} + 0.01\Delta p_{\text{s}} + 0.03\frac{{\partial p_{\text{cs}} }}{\partial t} - 0.014\varphi + 0.15v_{t}^{x} + 1.0,$$
(5)
$$x = 0.6\left( {1 - \frac{{\Delta p_{\text{s}} }}{215}} \right),$$
(6)
$$v_{\text{m}} = \left( {\frac{{100b_{\text{s}} }}{{\rho_{\text{ms}} }}\Delta p_{\text{s}} } \right)^{0.5}$$
(7)

where \(\Delta p_{\text{s}}\) in hPa, \(\frac{{\partial p_{\text{cs}} }}{\partial t}\) is the rate change of intensity in hPa/h, \(\varphi\) is the latitude in degrees and \(v_{\text{t}}\) is the cyclone translation speed in m/s.

The surface air density can be derived as:

$$\rho_{\text{s}} = \frac{{100p_{\text{s}} }}{{RT_{\text{vs}} }},$$
(8)
$$T_{\text{vs}} = \left( {T_{\text{s}} + 273.15} \right)\left( {1 + 0.61q_{\text{s}} } \right),$$
(9)
$$q_{\text{s}} = RH_{\text{s}} \left( {\frac{3.802}{{100p_{\text{s}} }}} \right)^{{e^{{\frac{{17.67T_{\text{s}} }}{{243.5 + T_{\text{s}} }}}} }} ,\; {\text{and}}$$
(10)
$$T_{\text{s}} = {\text{SST}} - 1,$$
(11)

where R = 286.9 J Kg−1 represents the gas constant for dry air, \(T_{\text{vs}}\) is the virtual surface temperature (in K), \(q_{\text{s}}\) is the surface moisture (in g Kg−1), \(T_{\text{s}}\) and SST are the surface temperature and sea surface temperature (both in °C), respectively, and \(RH_{\text{s}}\) is the surface relative humidity (assumed at 0.9 in the absence of direct observations). Detailed description and formulation is available in Holland et al. (2010).

3 Methodology

SMS (surface water modeling system) software is used to generate the unstructured triangular mesh used in the present study. General Bathymetric Chart of the Ocean (GEBCO) 30-arc second data is used as inland topography and ocean bathymetry owing to its high resolution. The generated mesh comprises 1,089,458 elements and 547,927 nodes. The mesh resolution varies from 100 m in the nearshore regions to about 30 km in the deeper waters. This selection of grid spacing is based on the study by Blain et al. (1998) and Rao et al. (2009). Their studies propose that a nearshore mesh resolution of 250 m or less is sufficient to represent the peak surge during cyclones. Along the landward boundary, the 15-m height contour is taken as a rigid boundary in order to compute the horizontal extent of inundation from the coastline, assuming that the storm surge height does not exceeded 15 m. Model mesh and bathymetry used for Hudhud and Phailin simulations are shown in Fig. 2a. Figure 2b, c shows the magnified view of the mesh around Konada (Andhra Pradesh) and Ganjam (Odisha) locations, respectively, in order to have a clear picture of fine grid spacing representing the coastal regions. The Le Provost tidal database (Le Provost et al. 1998) was used to prescribe the tidal constituents along the open ocean boundary. The 13 tidal constituents (K1, O1, P1, Q1, M2, N2, S2, L2, K2, T2, 2N2, NU2 and MU2) are provided along the open ocean boundary. The model was initially forced with the above-mentioned tidal constituents for a period of 61 days initiating with a cold start to establish steady-state conditions. Thereafter, the model-computed cold-start information was subsequently provided for model runs in a hot-start mode that additionally considers the wind forcing. The computed tide is also validated at Visakhapatnam and Paradip locations to ensure that the model performed well in simulating the water levels. A comparison of the water levels from the model against in situ observations at Visakhapattanam and Paradip locations is shown in the top and bottom panels of Fig. 3, respectively, which indicates a very good match. Both ADCIRC (in stand-alone mode) and coupled ADCIRC + SWAN models are used for simulations in the present study. The bottom friction coefficient was taken as 0.0028 (Zachry et al. 2013; Ziegler et al. 1993; Vongvisessomjai and Rojanakamthorn 1989), and the model time step prescribed as 4 s. The coupling time interval and SWAN time step were set to 600 s (Bhaskaran et al. 2013) which could capture well the nonlinear interaction effects arising due to currents and water level changes on the resultant wave field. The SWAN model is implemented in a non-stationary mode prescribed with 34 frequency bins and 36 directional bins. The frequency bins range between 0.04 and 1.0 Hz, and the wave direction has an angular resolution of 10°. Nonlinear wave–wave interaction in the deep water based on Hasselmann et al. (1985) is employed in the SWAN model. The bottom friction formulation used in the SWAN model is based on Madsen et al. (1988) which allows for the spatially varying roughness length. The constant quadratic bottom friction coefficient used in the study for both the coupled and uncoupled model is 0.0028 (Vongvisessomjai and Rojanakamthorn 1989; Ziegler et al. 1993; Zachry et al. 2013). The bottom friction coefficient selected in this study is best suited for the sandy bottom environment along the Andhra coast, an optimum configuration with both ADCIRC and SWAN models (Murty et al. 2014). However, the variable bottom friction option over the ocean and flooded/dried areas can further enhance the results. The primary and main forcing for the storm surge model is the cyclonic wind stress. In the present study, the wind and pressure fields are derived from the revised Holland dynamic wind model (Holland et al. 2010). The current version of ADCIRC used in the study applies the wind drag coefficient from Garratt (1977); SWAN applies a nearly identical wind drag coefficient from Wu (1982). The SWAN + ADCIRC model limits the wind drag coefficient to a maximum of 0.002. The wind drag can be better represented by considering the Powell sector-based wind drag formulation (Powell and Reinhold 2007). However, this will be the scope of a future study. Snapshots of the spatial wind field during the landfall time of Hudhud and Phailin are shown in Fig. 4a, b, respectively. The model-computed water level elevations, significant wave height, and inundation extents are also validated against the available field observations.

Fig. 2
figure 2

a Model mesh that shows the domain and bathymetry; b enlarged portion of the mesh showing the enhanced grid resolution in shallow waters around the Konada, Andhra Pradesh, region; c enlarged portion of the mesh showing the enhanced grid resolution in shallow waters around the Ganjam, Odisha, region

Full size image
Fig. 3
figure 3

Validation of modeled tide at Visakhapattanam (upper panel) and at Paradip (lower panel)

Full size image
Fig. 4
figure 4

Snapshot of spatial wind field during landfall of a Hudhud and b Phailin cyclones

Full size image

4 Results and Discussion

The ADCIRC model run in stand-alone mode considers the astronomical tidal forcing along the open-ocean boundary along with wind forcing, whereas the coupled ADCIRC + SWAN model also considers additional forcing due to the wave radiation stress. As the tide–surge interaction is quite significant in a shallow-water environment, the astronomical tide forcing forms an integral component and, therefore, must be included in the surge computations instead of linearly superposing the water level elevation after carrying out the storm surge computations. Owing to the importance of tide in altering the resultant water level/total water elevation (i.e., total water level obtained due to combined “surge and tide” from ADCIRC simulation and due to “surge, tide and wave” from coupled ADCIRC + SWAN simulation), the tidal component is included in the model simulations. However, the surge residual is also computed by removing the astronomical component from the resultant water level to evaluate the resultant water level changes solely attributed due to the wind forcing. The significance of a wave-induced setup in storm surge inundation modeling is examined based on simulations carried out for recent severe cyclones Hudhud and Phailin.

4.1 Cyclone Hudhud

Hudhud was an extremely severe cyclonic storm that made landfall near Visakhapatnam, ECI, on 12 October 2014 and also considered as the most intense cyclone that made landfall at this location during the last two decades. The track details of Hudhud cyclone are shown in Fig. 5. Model simulation (both coupled and uncoupled) was performed for 96 h. The H10-computed maximum wind speed was around 52 m/s (as shown in Fig. 4a) which is in close agreement to the IMD-reported value of 51 m/s (Kotal et al. 2015).

Fig. 5
figure 5

Tracks of Hudhud and Phailin cyclones

Full size image

Storm surge simulations were carried out using the ADCIRC stand-alone and coupled ADCIRC + SWAN models. In both these simulations, the computed peak surge was observed near Konada (83.57°E, 18.01°N), Vijayanagaram district. The uncoupled run showed a peak storm surge of about 2.4 m, whereas that for the coupled model run was about 3.0 m. Spatial representation of the total water elevation (hereafter TWE) and inundation extents computed by the uncoupled and coupled models are shown in Fig. 6a, b respectively. Figure 6c, d shows the magnified version seen in Fig. 6a, b, respectively, at the Konada location. Similarly, Fig. 6e, f shows the magnified versions corresponding to Fig. 6a, b, respectively, at Bontalakoduru (83.93°E, 18.21°N) location. It is very clear that the computed TWE and inundation extents from coupled runs are higher than those from the uncoupled runs. The maximum difference observed in the TWE between uncoupled and coupled model runs is about 0.6 m. This clearly demonstrates that the wave radiation stress in the coupled simulation contributes about a 20% increase in the resultant TWE. Spatial distribution of wave-induced water levels is shown in Fig. 7, which is the difference between TWEs from uncoupled and coupled model runs. Figure 7 clearly shows that the coastal stretch from the landfall point spanning up to Srikakulam is strongly influenced by wave-induced setup, with a peak value of 0.6 m corresponding to Cyclone Hudhud. Table 1 provides more details on the assessment of TWEs and inundation extent (with and without wave-induced setup) at different locations resulting from Hudhud landfall. The nearest available tide gauge station for validation of the computed surge residuals is located at Visakhapatnam (83.28°E, 17.68°N). A comparison of the computed TWE and surge residuals against measured data off Visakhapatnam is shown in Fig. 9a, b respectively. The comparison demonstrates a very good match between model computation and observation data. In addition, it is clear from Fig. 9b that the primary and secondary peaks of surge maxima are very well represented by the model, as seen in observations deciphering the fact that the modeled wind fields are realistic in estimating storm surge. Figure 9c and d illustrates the time evolution of TWE and surge residuals, respectively, at the location of peak surge (Konada) during Hudhud. The arrow mark shown in figure indicates the landfall time of Hudhud. As there was no observation available at the peak surge location, only the modeled TWE and surge residuals are shown in Fig. 9c, d, respectively. As seen from the figure, there is only a marginal difference in the surge residuals at Visakhapatnam from the coupled and uncoupled model runs. It clearly indicates that the role of wave-induced setup at this location is insignificant, as seen in the TWE. In contrast, at Konada, the wave-induced setup increased the storm surge heights as the cyclone approached the landfall location, attributed to diverse features in the coastal geomorphology. The results reveal that the effect of wave-induced setup on TWE depends on the nearshore dynamics and coastal geomorphology. In addition, the effect of wave-induced setup is also governed by the width of the continental shelf. A recent study attempted by Poulose and Rao (2017) reported on the influence of continental shelf geometry on the nonlinear interaction between storm surges, tides and wind waves based on an idealized experiment representing the west coast of India. Figure 8 shows the cross-shore bathymetry variation at Visakhapatnam and Konada locations. From this figure, it is very clear that the cross-shore bottom profile is flat near Konada (with steepness 1:600) as compared with the location off Visakhapatnam (1:200). This is reflected in the uncoupled and coupled model simulations, indicating a reduction in the root mean square error (RMSE) from 0.30 to 0.18 m, respectively. Figure 10 shows the comparison of significant wave height between in situ observation and modeling. Significant wave height observations during Hudhud was obtained from the Visakhapatnam wave rider buoy. The time stamp of significant wave heights represented along the abscissa in Fig. 10 correspond to the duration when Hudhud intensified from a severe cyclonic storm (SCS, October 10, 2014; 0300 UTC) to a very severe cyclonic storm (VSCS, October 10, 2014; 0900 UTC). The modeled significant wave heights are in good agreement with the observed wave heights. This agreement is also statistically significant, as indicated by the high correlation coefficient (CC) of 0.95 and RMSE of 0.9 m.

Fig. 6
figure 6

Spatial depiction of storm surge levels and computed inundation extents due to Hudhud by a uncoupled and b coupled runs. c Enlarged view of red-colored box in a at the Konada location, d enlarged view of red-colored box in b at the Konada location, e enlarged view of red-colored box in a at the Bontalakoduru location and f enlarged view of red-colored box in b at the Bontalakoduru location. (TWE is the acronym for total water elevation, and red arrows indicate the cyclone track)

Full size image
Fig. 7
figure 7

Spatial representation of wave-induced surge levels between Visakhapattanam and Srikakulam locations along the Andhra Pradesh coast. Black line with an arrow mark is the track of the cyclone)

Full size image
Fig. 8
figure 8

Bathymetric change towards off-shore at Visakhapattanam and Konada locations

Full size image
Table 1 Comparison of uncoupled- and coupled-model-computed surge heights and inundation extents along the coastal region affected by Hudhud
Full size table
Fig. 9
figure 9

Validation of computed a water levels (includes tide), b surge residuals at Visakhapattanam. Depiction of c computed water levels (includes tide), d surge residuals at Konada (peak surge locations)

Full size image
Fig. 10
figure 10

Validation of computed significant wave height at the Visakhapattanam wave rider buoy location

Full size image

4.2 Cyclone Phailin

Phailin was a very severe cyclonic storm that made landfall along the Odisha coast, India, on 12 October, 2013 and is regarded as the most intense cyclone after the Odisha super-cyclone (Kotal et al. 2015). Model simulation was carried out for 96 h since the cyclone developed over the open ocean. The track of Phailin is shown in Fig. 5. The coupled and uncoupled models computed the peak surge near the Ganjam (85.06°E, 19.37°N) location, Odisha. The storm surge computed with uncoupled model run was about 2.0 m, and the same using the coupled model was about 2.7 m, which shows the effect of wave radiation stress in the total water level elevation. Figure 11a, b illustrates the scenarios of storm surge and computed inundation extent by the uncoupled and coupled model runs, respectively. Figure 11c, d is the enlarged version corresponding to Fig. 11a, b at the Anandpur (85.54°E, 19.70°N) location. Similarly, Fig. 11e, f is the magnified version corresponding to Fig. 11a, b, respectively, for the Ganjam location. It is very clear that the computed storm surge heights and inundation extents are higher at both the locations with the coupled model run, with a difference of about 0.6 m. The study signifies that the contribution from wave radiation stress was about 30% in the total water level elevation. The spatial distribution of wave-induced water levels is shown in Fig. 12. It is clear from this figure that the entire coastal stretch spanning from landfall point to Srikakulam is affected by wave setup with a peak value of about 0.6 m during this event. The dark blue shade in the figure corresponds to the wave set-down process. Table 2 lists the comparison of storm surge levels and corresponding inundation extent as computed by the coupled and uncoupled models at different locations during this extreme event. It is clear from Table 2 that the model-computed inundation extents were close to observations in the coupled model runs. Comparison of computed TWE and surge residuals against observed data off the Paradip location are shown in Fig. 13a, b, respectively. The overall comparison showed a very good match between model and observed surge residuals. Figure 13c, d shows the time evolution of TWE and surge residuals, respectively, at the location of peak surge (Ganjam) during Phailin. The arrow marked in the figure indicates the landfall time. As there are no observational data available near the peak storm surge location, only the modeled TWE and surge residuals are shown in Fig. 13c, d. Statistical measures of comparison between uncoupled and coupled model runs indicate reduced RMSE from 0.14 to 0.09 m, proving the robustness of the coupled model.

Fig. 11
figure 11

Spatial depiction of storm surge and computed inundation extents due to Phailin by a uncoupled, b coupled simulations. c Enlarged view of the red-colored box in a at the Anandpur location, d enlarged view of red-colored box in b at the Anandpur location, e enlarged view of the red-colored box in a at the Ganjam location and f enlarged view of the red-colored box in b at the Ganjam location. (TWE is total water elevation, and the red arrow indicates the cyclone track)

Full size image
Fig. 12
figure 12

Spatial representation of wave-induced surge levels between Chatrapur and Puri locations along the Odisha coast (black line with arrow mark is the track of the cyclone)

Full size image
Table 2 Comparison of surge heights and inundation extents computed by uncoupled and coupled models along the coastal region affected by Phailin
Full size table
Fig. 13
figure 13

Validation of computed a water levels (includes tide), b surge residuals at Paradip. Depiction of c computed water levels (includes tide), d surge residuals at Ganjam (peak surge locations)

Full size image

The dynamic characteristics of still water level elevation, the wave-induced setup and the set-down processes along coastal areas, especially during an approaching storm, have increasing effects that depend on the mutual interaction between waves, surges, currents and tides (Murty et al. 2014). For both the Hudhud and Phailin cases, the coupled model simulation that included the wave–surge–tide interaction component produced greater surge heights and inundation extents than the uncoupled simulation. In the case of Hudhud, though the surge heights are greater in the case for the coupled versus uncoupled run, there is no significant difference in the extent of inundation. However, in the case of Phailin, a significant difference is found in both the surge heights and the inundation extent between coupled and uncoupled simulations. It is well known that variations in local topographical features can affect the extent of inland inundation. The flat topography surrounding the Ganjam location, where the peak surge occurred during this event, contributes to the observed differences. On the other hand, for the Visakhapatnam location, where Hudhud made its landfall, has a steep topographical gradient, thereby resisting the inland penetration of storm surge that was higher. It is noted from the study that incorporation of wave-induced setup escalates the inundation distance as seen for the Phailin case which in turn depends on the coastal geomorphologic features. It is also observed from this study that the inundation extent increased to a maximum by about 260 m and 50 m during the Phailin and Hudhud events, respectively. These observations are in line with the study reported by Wu et al. (2018). Figure 14 shows the comparison of significant wave height between observation and model output. Significant wave height data during Phailin was obtained from the Gopalpur wave rider buoy. The modeled significant wave heights are in good agreement with the observed significant wave heights. This agreement is also statistically significant, as indicated by the high CC of 0.96 and RMSE of 0.8 m.

Fig. 14
figure 14

Validation of computed significant wave height at the Visakhapattanam wave rider buoy location

Full size image

5 Conclusions

Previous studies have shown that wave–current interactions can significantly influence coastal currents and water level elevations. From a global perspective, very few studies are available for other coastal regions, and there are no studies for the Indian coast that examined the overall contribution of wave setup on storm surge-induced coastal inundation. The present study is an application of uncoupled ADCIRC and a coupled ADCIRC + SWAN models to simulate storm surges and associated coastal inundation for very severe cyclonic storms Hudhud and Phailin cases. To explore the influence of surface gravity waves on storm surge and coastal inundation, a series of experiments were conducted. While the interaction between tide and surge and their mutual effects during extreme events such as tropical storms have been studied previously for the ECI, a detailed investigation of the interaction mechanism between tide, wave and surge, and the manner in which it affects storm surge heights and coastal inundation, has been lacking. The inclusion of wave radiation stress is very critical for improving model performance and for better representation of computed storm surge heights (Araújo et al. 2011). In the present study, the contribution of wave radiation stress and its influence on surge heights and coastal inundation was investigated during the recent very severe cyclonic storms Hudhud and Phailin that made landfall along the ECI. A finite element-based unstructured model, ADCIRC, in a stand-alone mode, and the ADCIRC + SWAN model (coupled mode) were used to perform the simulations. Meteorological forcing fields were generated using the Holland-2010 formulation to force the storm surge model. Results showed that the wave-induced radiation stresses contributed an approximately 20% (30%) increase in surge heights at the peak surge locations during Hudhud (Phailin) events, supported by Dietrich et al. (2011) and Sheng et al. (2010). From the time-series distribution for the Hudhud case, it is evident that the wave setup effect on total water elevation off the Visakhapatnam location is less significant or negligible. The results also indicate that the coupled model simulation showed an increase in the inundation extent along the Odisha coast during the Phailin event. On the other hand, there was no significant difference observed in the inundation extent with and without wave setup for the Hudhud simulation, indicating that local nearshore topography plays a significant role in governing the horizontal extent of inundation even though the surge heights are large. The study further signifies that the effect of wave–current interaction must necessarily be taken into account in storm surge calculations and, more importantly, in better representation of the inundation extent. It brings to light the importance of flat topography in modulating the extent of inundation and the role of wave radiation stress in hydrodynamic modeling. Based on this study, it is clear that the coupled modeling system performed significantly better, proving its usefulness for real-time forecasting of storm surge and associated flooding.