## 1. Introduction

Coastal areas are exposed to a range of natural hazards, including flooding due to storm tide and wave overtopping. In low-lying developed areas, coastal flooding can cause negative social, economic, and environmental impacts. For example, Hurricane Sandy in 2012 caused intense flooding along the northeastern coast of the United States, especially in and around the New York City, New York, metropolitan area, resulting to at least $50 billion in economic losses and 159 direct and indirect deaths (Blake et al. 2013). To effectively minimize flood risks, flood forecasting and warning systems must provide accurate flood information to emergency managers and decision-makers. Producing accurate flood forecast requires the computational models to properly take into account the important physical processes that affect the total water elevation and, consequently, the flooded areas.

In coastal regions affected by storms, the *storm tide* comprises the normal astronomical tide plus a tidal departure, known as *storm surge*, due to the combined hydrodynamic effects of a passing storm (National Ocean Service 2016). The influence of storm surges and astronomical tides on the storm tide at the coast can be simulated using a coastal ocean circulation model based on the three-dimensional (3D) Navier–Stokes equations or their depth-integrated derivatives (e.g., Blumberg and Mellor 1980; Luettich et al. 1992; Shchepetkin and McWilliams 2005; Chen et al. 2006). Locally, the storm tide is measured as the observed water level averaged over a period of 3 min, to filter out the random effects of surface waves on the water level. Waves during a storm can cause wave-induced setup/setdown, that is, an observable mean water level change due to momentum transfer from breaking waves into the water column. A phase-averaged wave model is a suitable tool to simulate the initiation, propagation, and dissipation of surface waves (e.g., Booij et al. 1999; Mellor et al. 2008, ; Massey et al. 2011; Donelan et al. 2012). Given that wave setup/setdown can contribute to storm surge, accurate predictions of the water elevation and, consequently, the flooded areas require a coupled circulation–wave model that properly takes into account wave–current interactions, especially due to wave momentum flux commonly known as radiation stress, wave-induced bottom friction, and wave-induced water surface roughness.

In recent years, several coastal ocean circulation models have been coupled with surface wave models, especially third-generation wave models, such as the Simulating Waves Nearshore (SWAN) model. For example, Allard et al. (2014) used the Earth System Modeling Framework (ESMF) to couple the Navy Coastal Ocean Model (NCOM) and SWAN. They modified NCOM to take into account the depth-independent wave radiation stress based on Longuet‐Higgins and Stewart (1964), the wave-enhanced bottom stress based on the methods of Signell et al. (1990) and Davies and Lawrence (1994), the Stokes drift current, and the vertical mixing due to Langmuir turbulence. Roland et al. (2012) coupled the Semi-Implicit Eulerian–Lagrangian Finite Element (SELFE) model and the spectral Wind Wave Model II (WWMII). Their model considered the wave–current interactions through the depth-independent wave radiation stress developed by Longuet-Higgins and Stewart (1962, 1964), the surface stress based on the theory of Janssen (1991), and the wave-induced bottom stress based on the theories of Grant and Madsen (1979) and Mathisen and Madsen (1996). Dietrich et al. (2011) coupled SWAN and the Advanced Circulation (ADCIRC) model, a vertically integrated circulation model. The influence of waves on the mean flow was implemented, as in all of the abovementioned models, using the depth-independent radiation stress of Longuet-Higgins and Stewart (1964) and Battjes (1972). Warner et al. (2008) coupled the Regional Ocean Modeling System (ROMS) and SWAN, and Uchiyama et al. (2010) implemented wave–current interactions with a vortex-force formalism in ROMS. Moon (2005) coupled the Princeton Ocean Model (POM) and the wave model WAVEWATCH II.

The high computational cost of full-spectrum wave models makes most of the existing coupled circulation–wave models, such as those cited above, suboptimal for direct atmosphere–wave–current coupling in ensemble flood forecasting systems (EFSs) and probabilistic flood hazard assessments. EFSs attempt to generate a representative future water level based on multiple numerical predictions conducted using perturbed physical parameters and initial conditions (e.g., Zou et al. 2013). Operational ensemble flood forecast systems require over a 100 simulations every forecast cycle to robustly address worst-case scenario metrics like the 10% chance exceedance flood (e.g., Forbes et al. 2014). These systems are increasingly being utilized because they are more valuable for decision support than deterministic forecasting systems. For instance, the Stevens Flood Advisory System (SFAS) uses 120 ensemble members to provide information about meteorological and oceanographic conditions in real time as well as forecasts up to 4 days out (Georgas et al. 2016; http://stevens.edu/SFAS). The Probabilistic Hurricane Storm Surge (P-Surge) model of the National Oceanic and Atmospheric Administration (NOAA) creates many statistically probable storms to get a better idea of the storm surge within the next 80 h (http://slosh.nws.noaa.gov/psurge2.0/; Taylor and Glahn 2008). However, because of the high computational cost of full-spectrum wave models, the currently available EFSs are based on a stand-alone circulation model or, at best, a circulation model coupled with a simple wave model that neglects important wave physical processes, such as wave setup. Similar to EFSs, probabilistic flood hazard assessments require many, ideally rapid, storm simulations. These studies seek to characterize a region’s flood probabilities based on simulations of thousands of storm events that represent a climatology of all possible storms and tides. For example, Orton et al. (2016) simulated 1557 events to assess flood hazards for New York Harbor. Access to computationally cost-effective circulation–wave models is useful for both flood hazard assessments and ensemble forecasting systems.

Mellor, Donelan, and Oey (2008) developed a computationally cost-effective spectral wave model, appropriate for coupling with 3D circulation models. To reduce the computational expenses, MDO uses a specified spectrum shape to parameterize the wave–wave interaction processes and the energy distribution in the frequency space. Thus, the model solves the spectral equation only in the geographical and directional spaces. Mellor et al. (2008) evaluated their stand-alone wave model using laboratory and field data. The model results for Hurricane Katrina showed that the accuracy of the modeled wave properties is comparable to buoy data and the results produced by a third-generation wave model.

In the present study we couple the Stevens Estuarine and Coastal Ocean Model (sECOM) and the MDO wave model. sECOM and MDO are two-way coupled through water depth and velocity, wave-induced vertically resolved radiation stress, and wave-enhanced bottom and water surface stresses. Compared to an older-generation momentum balance wave model based on Donelan (1977), as adopted and two-way-coupled to sECOM in the past (Georgas et al. 2007) and used in the New York Harbor Observing and Prediction System (Georgas and Blumberg 2010), the two-way coupling of MDO to sECOM provides more comprehensive spectral wave physics, swell propagation, closer coupling and, importantly, adds wave-induced radiation stress. Results of the coupled circulation–wave model are compared with existing laboratory measurements and the data observed during Hurricane Sandy in New York–New Jersey (NY–NJ) Harbor. The subsequent sections of this paper include the description of the numerical models, the coupling process, the model results and discussion, and the summary and conclusions.

## 2. Coupled circulation–wave model

### a. Wave model

The MDO wave model developed by Mellor et al. (2008) calculates the wave properties by solving the wave energy balance equation in geographical and directional spaces. The core of MDO is the spectrum of Donelan et al. (1985; the DHHS spectrum), which is based on extensive wind-wave data obtained on Lake Ontario and does contain elements of the Joint North Sea Wave Project (JONSWAP) spectrum (Hasselmann et al. 1973). If the total wind-driven energy, wind speed, and peak frequency are known, then the spectrum over the entire frequency is determined. An integral of the DHHS spectrum over frequency and wind direction provides a relation between total wind-wave energy, wind speed, and peak frequency of the wind-driven portion of wave direction; thus, aside from wind speed, the dependent variables are total wind-wave energy and wave propagation direction.

*f*based on the DHHS spectrum, making it computationally cost effective. The directionally dependent wave energy equation in a sigma coordinate system is

*E*

_{θ}is the frequency integrated energy;

*t*is the time;

*x*

_{i}is the horizontal coordinate (

*i*=

*x*,

*y*and

*j = x*,

*y*);

*c*

_{gi}is the group speed;

*u*

_{Ai}is the Doppler (advective) velocity formulated in Mellor (2003, 2008);

*c*

_{θ}is the refraction speed, which is calculated based on Komen et al. (1994) and Golding (1978);

*D*=

*h*+

*η*is the total water depth, where

*h*is the bathymetry referenced to a vertical datum and

*η*is the mean (phase averaged) surface elevation referenced to the same datum;

*S*

_{r,ij}is the wave radiation stress;

*U*

_{i}is the horizontal velocity representing current plus the Stokes drift; ς is the sigma variable (−1<

*ς*< 0) defined as

*ς*= (

*z*−

*η*)/

*D*, where

*z*is the vertical coordinate;

*S*

_{θ,in}is the wind growth source term; and

*S*

_{θ,Sdis}and

*S*

_{θ,Bdis}are the dissipation terms due to wave processes at the surface and bottom, respectively. The dissipation terms take into account all important wave energy dissipation processes, such as whitecapping and depth-induced breaking (more details in MDO). The group speed, refraction speed, and Doppler velocity are energy-weighted quantities averaged over frequency. The frequency-integrated energy

*E*

_{θ}is a function of spatial coordinates and time and, importantly, a function of wave direction

*θ*, where −π <

*θ*≤ π. Peak frequency is also a function of

*θ*and is determined by an equation derived from wavenumber irrotationality and conservation of wave crests. It contains an empirical source term that drives the frequency toward wind-driven peak frequency in regions of

*θ*, which are wind driven and determined from the DHHS spectrum given the wind direction. Otherwise, regions that are not wind driven are considered to be swell.

*k*is the wavenumber,

*δ*is the Kronecker delta, and

MDO solves the energy equation [Eq. (1)] on an Arakawa C, orthogonal curvilinear grid with a terrain-following (sigma) vertical coordinate. The circulation model, described in the following section, adopts a similar computational grid, making the coupling process convenient.

### b. Circulation model

The circulation model, sECOM, is a variant of POM, originally developed by Blumberg and Mellor (1980, 1987). The model has been improved over time and successfully applied to oceanic, coastal, and estuarine waters (e.g., Oey et al. 1985; Galperin and Mellor 1990; Blumberg et al. 1999, 2015; Blumberg and Georgas 2008; Georgas and Blumberg 2010; Georgas 2012; Orton et al. 2012, 2015, 2016; Marsooli et al. 2016). sECOM, in its parallel message passing interface (MPI) version (Jordi et al. 2017), is also the 3D circulation model of SFAS, an ensemble flood forecasting resource for emergency preparedness for the U.S. mid-Atlantic and Northeast (Georgas et al. 2016). Because the MPI version of sECOM was not available during the present study and, moreover, the MPI version of MDO’s source code has not been developed yet, we use the serial version of sECOM compiled using an Open Multi-Processing (OpenMP) autoparallel compiler.

*U*

_{i}is the combined current plus the Stokes drift, whereas Ω is the sigma coordinate (nearly) vertical velocity such that

*w*is the Cartesian vertical component of velocity and ∇ is the horizontal gradient operator;

*g*is the acceleration due to gravity;

*p*

_{atm}is the atmospheric pressure;

*p*is the hydrostatic pressure;

*ρ*is the water density;

*ρ*

_{0}is a reference density;

**F**

_{c}is the Coriolis force vector [−

*fυ*,

*fu*] and

*f*is the Coriolis parameter;

*τ*

_{T,i}is the turbulent stress defined as

*K*

_{M}(∂

*U*

_{i}/∂

*ς*), where

*K*

_{M}is the vertical eddy viscosity; and

**F**

_{H}represents the horizontal diffusion terms determined from the horizontal eddy viscosity

*A*

_{M}and velocity gradients. Term

*K*

_{M}is calculated using the Mellor–Yamada level 2.5 turbulence closure model (Mellor and Yamada 1982). Term

*A*

_{M}, which parameterizes all horizontal transport processes not resolved by the model grid, is calculated based on the Smagorinsky parameterization method (Smagorinsky 1963; Blumberg and Mellor 1987).

The third term on the left-hand side of Eq. (5) takes into account the momentum transfer from breaking waves into the mean flow through the depth-dependent wave radiation stress theory. Theoretical support for the radiation stress theory is provided by Mellor (2016). The implementation of this theory into coastal ocean models has previously shown that it correctly captures the momentum transfer from breaking waves into the mean flow (e.g., Xia et al. 2004; Liu and Xie 2009; Wu 2014). For example, Sheng and Liu (2011) implemented three different radiation stress formulations into their coupled circulation–wave model and concluded that the depth-dependent radiation stress of Mellor (2008) accurately captures mean currents with correct direction inside and outside the surfzone.

*τ*

_{s}= 0.5

*ρ*

_{a}

*C*

_{D,wind}(

*U*

_{10})

^{2}, where

*ρ*

_{a}is the air density and

*U*

_{10}is the wind speed at 10-m height. The logarithmic law can be used to compute the wind drag coefficient as

*C*

_{D,wind}= 2[

*κ*/ln(10/

*z*

_{0,wind})]

^{2}, where

*z*

_{0,wind}is the water surface roughness length. The original version of sECOM estimates

*z*

_{0,wind}based on the height and steepness of surface waves, using the empirical formula of Taylor and Yelland (2001). MDO adopts the wave-age-dependent formula of Donelan (1990) to calculate

*z*

_{0,wind}and, in turn,

*S*

_{θ,in}in Eq. (1). To be consistent with the wave model, we also adopt the same wave-age-dependent formula in sECOM. Thus,

*z*

_{0,wind}is calculated from Donelan (1990) as

*f*

_{p}is the peak wave frequency. To accommodate large wind velocities, we limit

*C*

_{D,wind}to 0.003 (Powell et al. 2003).

*τ*

_{b}= 0.5

*ρC*

_{D,b}(

*U*

_{b})

^{2}, where

*U*

_{b}= (

*u*

_{b}

^{2 }+

*υ*

_{b}

^{2})

^{0.5}is the near-bottom mean current,

*u*

_{b}and

*υ*

_{b}are the horizontal components of the near-bottom velocity, and

*C*

_{D,b}is the bottom drag coefficient. In the absence of surface waves,

*C*

_{D,b}is estimated based on the law of the wall as

*κ*= 0.41 is von Kármán’s constant,

*δ*

_{ς,b}is the thickness of the bottom sigma layer, and

*z*

_{0}is the bottom roughness length. In the absence of bed forms (e.g., ripples),

*z*

_{0}is estimated as 2.5

*D*

_{50}/30, where

*D*

_{50}is the median grain size. The

*C*

_{D,min}option sets of a floor value for deeper water columns, where the law of the wall scaling is not applicable across the bottom sigma layer. In this study,

*C*

_{D,min}is set to 0.001. In the presence of surface waves, the bottom drag coefficient is estimated as

*C*

_{cw}is an effective wave–current friction coefficient that is calculated based on the theory of Grant and Madsen (1979) and in a form similar to Signell et al. (1990). The effective friction coefficient can be computed through an iterative process described in the appendix.

sECOM solves the governing equations using the finite-difference method on an Arakawa C grid with a terrain-following (sigma) vertical coordinate and an orthogonal curvilinear horizontal coordinate. The model adopts a mode-splitting technique (Simons 1974; Madala and Piacsek 1977) to separate the external gravity (barotropic) mode from the internal gravity (baroclinic) mode. The external mode equations, which are obtained by integrating the continuity and momentum Eqs. (4)–(6) over the water depth, must be solved for a sufficiently small time step (*dt*_{ext}) in order to accurately capture the fast-moving external gravity waves. The internal mode Eqs. (4)–(6) are computationally more expensive than the external mode equations but can be solved at much larger time steps (*dt*_{int}) due to the slow speed of internal gravity waves. Thus, the model solves the external mode with a smaller time step than the internal mode. In the present study, the external time step is 10 times smaller than the internal time step, that is, *dt*_{int} = 10 *dt*_{ext}.

### c. Model coupling

sECOM is coupled with the MDO wave model at the source code level. sECOM and MDO share the same horizontal computational grid, which makes the data exchange between them convenient and efficient. sECOM’s main program calls the wave model within time steps of *ndt*_{int}, where *n* is a predefined whole number equal or greater than one. The time step of the wave model is a fraction of the internal mode time step, that is, *dt*_{wave} = *ndt*_{int}*/m*. In this study, we set *n* = 1 and *m* = 2.

Water elevation and velocity fields are the feedbacks of the circulation model to the wave model. Water elevation variations, especially in shallow regions and during storm events, can significantly affect the water depth. A changing water depth can, in turn, impact the wave energy dissipation due to depth-induced breaking (Battjes and Janssen 1978) and bottom friction. In addition to water depth, the velocity field may also greatly impact the results of the wave model. For example, surface currents alter the relative wind speed (i.e., wind speed relative to surface current) and, in turn, *S*_{θ,in} on the right-hand side of the wave energy balance [Eq. (1)]. Currents also impact current-induced refraction and energy exchange with the mean flow velocity, the third and fourth terms on the left-hand side of Eq. (1), respectively.

The wave model feeds back to the circulation model wave properties, such as wave height, length, and frequency, to estimate the water surface roughness and the bottom friction for use in wind and bottom stress computations. Wave properties are also used for the calculation of radiation stress gradient terms in the momentum equations [Eqs. (5)]. These gradient terms take into account the transfer of momentum fluxes from breaking waves into the water column and determine the wave setup/setdown.

### d. Model evaluation

The coupled model is evaluated using laboratory and field data. We first evaluate the model using measurements from three laboratory experiments reported in the literature. The model is then evaluated using measurements in NY–NJ harbor during Hurricane Sandy. The next section presents the results and discussion for each case study.

## 3. Results and discussion

### a. Wave height and setup over an idealized beach

We first evaluate the coupled sECOM–MDO model using laboratory data measured by Stive (1985). The laboratory experiment was conducted in a large-scale wave tank—233 m long, 5 m wide, and 7 m deep—at the Delft Hydraulics’ Delta Flume, the Netherlands. The flume was built of a horizontal section followed by a sloping beach with a constant slope of 1:40. The flume bed consisted of fine sand with a median grain diameter of 0.22 mm. A piston-type wave board generated random waves with a significant wave height of 1.41 m and a peak wave period *T*_{p} of 5.41 s. The still water depth in the horizontal section of the flume was 4.1 m. We use the measured longitudinal profiles of wave height and water surface elevation to evaluate the performance of the coupled sECOM–MDO model.

The model is run on a computational mesh with a uniform horizontal grid spacing of 1 m and 16 evenly distributed vertical layers. The computational time step *dt*_{int} is set to 0.05 s. Term *z*_{0} is equal to 2 × 10^{−5} m, which is calculated based on the median grain diameter. The simulation period is 30 min, sufficient for the model to reach the steady-state solution. The measured wave properties and water elevation are imposed at the boundary of the model.

Figure 1 compares the longitudinal profiles of measured and calculated significant wave height and water surface elevation. The model accurately simulates the wave height reduction, occurring mainly due to wave breaking on the sloping beach. The model also well produces the water surface elevation. The measured and calculated water surface profiles show that, first, the water level decreases slightly (Phillips 1977) and then rises within the wave breaking zone, mainly due to the radiation stress gradient terms in the momentum equations. The longitudinal profiles of water surface elevation shown in Fig. 1 indicate that the model accurately captures the peak setup, which is about 0.06 m, whereas it slightly underestimates the spatial gradients of wave setup. The modeled water surface profile shows a milder slope compared to the slope of the measured profile. This is not caused by inadequate resolution, as we find there is no change when using a finer resolution model grid. The underprediction of spatial wave setup gradients may arise because the model does not resolve all physical processes that influence wave–current interactions in surf- and swash zones. For example, the assumption of hydrostatic pressure in the circulation model may not be valid in the wave breaking zone. More accurate results might be achieved by using a nonhydrostatic model such as the 3D volume of fluid (VOF) and phase-resolving hydrodynamic models (e.g., Marsooli and Wu 2014a,b). However, the nonhydrostatic models are computationally expensive even for laboratory-scale case studies. Other sources of discrepancy in water surface gradients can be due to errors related to the turbulence closure model, model parameters, and numerical methods used in the circulation and wave models.

### b. Wave height and setup over an uneven beach

The coupled model is also evaluated using laboratory experiments of Bores (1996) to investigate the performance of the model to simulate wave-induced water elevation over an irregular beach. Bores (1996) conducted a series of laboratory experiments in a 40-m-long, 0.8-m-wide, and 1.08-m-high wave flume located at the Fluid Mechanics Laboratory of the Delft University of Technology, the Netherlands. A sandbar was built within the wave breaking zone of the beach (Fig. 2, bottom plots). The fixed bottom profile of the flume was made of sand with a smoothed concrete top layer. The experiments were carried out under three different wave conditions. We compare the results from the coupled sECOM–MDO model with measurements from the most moderate and the most extreme wave conditions, that is, waves with a significant wave height of 0.1 m and a peak wave period of 3.4 s (experiment 1C) and also a height of 0.22 m and a peak period of 2.1 s (experiment 1B).

Comparisons between measured (circles) and calculated (solid lines) (top) *H*_{s} and (middle) *η* for the laboratory experiments of Bores (1996). (left) Experiment 1C (*H*_{s} = 0.1 m and *T*_{p} = 3.4 s) and (right) experiment 1B (*H*_{s} = 0.22 m and *T*_{p} = 2.1 s), and (bottom) the beach profile (the origin of the *x* axis is located at the toe of the beach).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Comparisons between measured (circles) and calculated (solid lines) (top) *H*_{s} and (middle) *η* for the laboratory experiments of Bores (1996). (left) Experiment 1C (*H*_{s} = 0.1 m and *T*_{p} = 3.4 s) and (right) experiment 1B (*H*_{s} = 0.22 m and *T*_{p} = 2.1 s), and (bottom) the beach profile (the origin of the *x* axis is located at the toe of the beach).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Comparisons between measured (circles) and calculated (solid lines) (top) *H*_{s} and (middle) *η* for the laboratory experiments of Bores (1996). (left) Experiment 1C (*H*_{s} = 0.1 m and *T*_{p} = 3.4 s) and (right) experiment 1B (*H*_{s} = 0.22 m and *T*_{p} = 2.1 s), and (bottom) the beach profile (the origin of the *x* axis is located at the toe of the beach).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

The computational grid consists of a horizontal grid spacing of 0.1 m and 16 vertical layers. The total simulation time is 30 min and *dt*_{int} = 0.005 s. The bottom roughness length is 2 × 10^{−5} m. The model boundary conditions are defined based on the measured wave properties and water elevations.

Figure 2 compares the measured and calculated significant wave heights and water surface elevations. Although the model overestimates the water elevation over the sandbar for experiment 1B, the overall agreements between the measured and calculated profiles over the two experiments are quite good. Roland et al. (2012) also used this laboratory experiment to evaluate their coupled circulation–wave model (SELFE–WWMII). Their results (using the default parameterization for triads and wave breaking) showed underestimated wave heights and overestimated water surface elevations within the wave breaking zone, which may be due to the depth-averaged radiation stresses adopted in their model, whereas radiation stresses in coupled sECOM–MDO model are depth dependent as in Mellor (2003, 2015).

### c. Wave-induced undertow current

We use the data measured during the large-scale laboratory experiment of the European Large Installation Plan (LIP; Roelvink and Reniers 1995) to evaluate the coupled circulation–wave model for wave-induced undertow currents. The LIP laboratory wave tank was 233 m long, 5 m wide, and 7 m deep. The beach profile, shown in Fig. 3, was made of sand with a median grain diameter of 0.22 mm. Experimental runs were conducted under different incident wave conditions, including slightly erosive, highly erosive, and strongly accretive wave conditions. We compare the model results with measurements from experiment 1A, which was carried out under slightly erosive, narrow-banded random waves with a significant wave height of 0.9 m, a wave period of 5 s, and a still water depth of 4.1 m.

Comparisons between measured (circles) and calculated (solid lines) (top) *H*_{s} and (middle) *η* for the laboratory experiment 1A of Roelvink and Reniers (1995), and (bottom) the beach profile.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Comparisons between measured (circles) and calculated (solid lines) (top) *H*_{s} and (middle) *η* for the laboratory experiment 1A of Roelvink and Reniers (1995), and (bottom) the beach profile.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Comparisons between measured (circles) and calculated (solid lines) (top) *H*_{s} and (middle) *η* for the laboratory experiment 1A of Roelvink and Reniers (1995), and (bottom) the beach profile.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

The computational domain consists of 20 vertical layers, and the horizontal grid spacing is 1 m. The total simulation time is 60 min and *dt*_{int} is 0.01 s. The bottom roughness length is set to 2 × 10^{−5} m. The measured water surface elevation and wave properties offshore are used as the model boundary conditions.

Figure 3 shows the measured and calculated profiles of the wave height and the water surface elevation. The wave height slightly increases due to depth-induced shoaling and then reduces as waves propagate within the wave breaking zone. The measured and calculated water surface elevations demonstrate wave setup in the shallow region of the domain. The breaking waves in this region transfer their momentum into the water column, leading to an increase in the water surface elevation. While the water surface gradient is slightly underestimated, the peak wave setup is matched in the nearshore area.

Figure 4 shows the measured and calculated vertical distributions of the streamwise velocity. The coupled sECOM–MDO model favorably captures the vertical distribution of wave-induced currents. However, the model overestimates the magnitude and vertical gradients of undertow currents at deeper regions of the domain. Both measurements and model results demonstrate that the impact of waves on water velocities is small in deeper water, for example, at *x* = 65 m, where *x* is the distance from the upwave end of the wave tank. In contrast, wave-induced flow velocities become important with distance shoreward from *x* = 65 m. As waves enter the shallower region, they break and transfer their momentum into mean momentum preferentially in the surface and, consequently, generate the undertow current. The current moves offshore into the lower part of the water column and onshore into the upper part. The model captures the zero crossing in the velocity distribution in the water column.

Comparisons between measured (circles) and calculated (solid lines) vertical profiles of the wave-induced streamwise velocity *u* (negative values directed offshore) for the laboratory experiment 1A of Roelvink and Reniers (1995).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Comparisons between measured (circles) and calculated (solid lines) vertical profiles of the wave-induced streamwise velocity *u* (negative values directed offshore) for the laboratory experiment 1A of Roelvink and Reniers (1995).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Comparisons between measured (circles) and calculated (solid lines) vertical profiles of the wave-induced streamwise velocity *u* (negative values directed offshore) for the laboratory experiment 1A of Roelvink and Reniers (1995).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

### d. Storm tides and waves generated by Hurricane Sandy in NY–NJ harbor

During 22–29 October 2012, Hurricane Sandy swept through the Caribbean and up the East Coast of the United States with the track shown in Fig. 5. Sandy made landfall in the United States as a posttropical cyclone at about 2330 UTC near Brigantine, New Jersey, with a minimum central pressure of 945 hPa and maximum sustained winds of 36 m s^{−1} estimated by the National Hurricane Center (NHC). Sandy drove damaging waves and storm surges into the New York and New Jersey coastlines, especially in and around the New York City metropolitan area. Based on the NHC’s Tropical Cyclone Report (Blake et al. 2013), Hurricane Sandy caused at least $50 billion in damage in the United States, about $2.88 billion in damage in the Caribbean, and at least 147 direct deaths across the Atlantic basin with 72 of these fatalities in the mid-Atlantic and northeastern United States. Since Hurricane Agnes in 1972, Sandy caused the greatest number of the U.S. direct fatalities related to a tropical cyclone outside of the southern states.

Track of Hurricane Sandy based on the data from NOAA’s NHC. Sandy made landfalls on Jamaica (24 Oct), Cuba (25 Oct), and the United States (29 Oct).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Track of Hurricane Sandy based on the data from NOAA’s NHC. Sandy made landfalls on Jamaica (24 Oct), Cuba (25 Oct), and the United States (29 Oct).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Track of Hurricane Sandy based on the data from NOAA’s NHC. Sandy made landfalls on Jamaica (24 Oct), Cuba (25 Oct), and the United States (29 Oct).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

To evaluate the performance of the coupled sECOM–MDO model for coastal-scale problems, we implement the model to hindcast storm tides and waves generated by Hurricane Sandy in NY–NJ harbor. The computational grid, shown in Fig. 6, is the high-resolution curvilinear grid of the New York Harbor Observing and Prediction System (NYHOPS). The computational grid consists of 147 × 452 nodes and its resolution ranges from approximately 7.5 km at the open ocean boundary to less than 50 m in NY–NJ harbor (Georgas and Blumberg 2010). The grid consists of 10 vertical layers. The NYHOPS grid is nested in the regional-scale Stevens Northwest Atlantic Prediction (SNAP) model domain (Blumberg et al. 2015), allowing us to determine the boundary conditions for use in sECOM–MDO model. A 2012 mean sea level of 0.08 m and climatological seasonal sea level for 30 October of 0.03 m were used as sea level offsets, applied with tides as offshore boundary conditions for the simulation. Reanalysis data from the Oceanweather Inc. (www.oceanweather.com) are used to compute atmospheric forcing due to wind, sea level pressure, and heat flux. The method of Ahsan and Blumberg (1999; see also Bhushan et al. 2010) is used to compute the heat fluxes, including shortwave solar and longwave atmospheric radiations, and sensible and latent heat fluxes. Freshwater inputs are based on data from the U.S. Geological Survey (USGS) and the U.S. Environmental Protection Agency (EPA). The geographical locations of freshwater inputs are similar to those in NYHOPS [see more information in Georgas and Blumberg (2010) and Georgas et al. (2016)].

(a) Computational grid and bathymetry of NYHOPS domain. (b) Location of water elevation sites (squares) and buoy stations (triangles) in NY–NJ harbor and the apex of the New York Bight, where the observed data during Hurricane Sandy are used to evaluate sECOM–MDO. (c) Key locations in NY–NJ harbor.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

(a) Computational grid and bathymetry of NYHOPS domain. (b) Location of water elevation sites (squares) and buoy stations (triangles) in NY–NJ harbor and the apex of the New York Bight, where the observed data during Hurricane Sandy are used to evaluate sECOM–MDO. (c) Key locations in NY–NJ harbor.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

(a) Computational grid and bathymetry of NYHOPS domain. (b) Location of water elevation sites (squares) and buoy stations (triangles) in NY–NJ harbor and the apex of the New York Bight, where the observed data during Hurricane Sandy are used to evaluate sECOM–MDO. (c) Key locations in NY–NJ harbor.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Term *dt*_{int} is 10 s. Term *z*_{0} is calibrated to 5 × 10^{−4} m. The calibration was carried out by comparing the measured water elevations with the calculated results from models with different bottom roughness lengths while other model parameters were constant. The wave model discretizes the directional space into 24 bins (15°). The model with more directional bins did not much affect the model results but increased the computational time. The sensitivity analysis carried out by Gibbs et al. (2012) for the Gulf of Mexico also showed that increasing the resolution from 24 to 36 bins in SWAN did not have much effect on their results but led to higher computational cost.

To evaluate the model performance, we compare the model results with observations at three water elevation sites and two buoy stations. The model results are compared to significant wave heights and average wave periods observed at buoy stations 44025 and 44065 in the apex of the New York Bight (Fig. 6). These buoys are owned and maintained by NOAA’s National Data Buoy Center (NDBC). The model results are also compared to water elevations recorded at the Battery (New York), Sandy Hook (New Jersey), and Staten Island (New York; RIC-001WV). The Battery and Sandy Hook sites are permanent coastal tide gauges owned and maintained by NOAA. The Battery site is located near Battery Park in Lower Manhattan, New York, and the Sandy Hook tidal gauge is located near the mouth of NY–NJ harbor. The other water elevation site, RIC-001WV, was a temporary gauge deployed by USGS. The gauge was located at Lower New York Bay, Staten Island. The USGS gauge recorded the pressure-induced water level variations. We applied a running average (6-min interval) on the data to remove wave-induced fluctuations and to generate the water elevation time series.

*X*is the variable being compared, the overbar means time mean, and

*N*is the number of data points. Perfect agreement between the model results and observations gives RMSE and bias values of zero and a skill of one. The maximum RMSEs in the calculated significant wave height and average wave period are 0.72 m and 1.48 s, respectively. Discrepancies between the model results and observations can be due to insufficient grid resolution, errors in measurements and meteorological data, errors in empirical formulas and numerical methods used in the circulation and wave models, and the complexity of modeling hurricane events where, for example, the wind field changes rapidly over multiple scales.

Observed (solid lines) and calculated (dashed lines) time series of *H*_{s} and average wave period *T* induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (solid lines) and calculated (dashed lines) time series of *H*_{s} and average wave period *T* induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (solid lines) and calculated (dashed lines) time series of *H*_{s} and average wave period *T* induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Statistical assessment of the modeled *H*_{s} and average *T* for Hurricane Sandy between 28 and 31 Oct 2012.

Time series of observed and calculated total water elevations (storm tides) *η*_{t} relative to the North American Vertical Datum of 1988 (NAVD88) are shown in Fig. 8. Though the peak storm tide at RIC-001WV is slightly overestimated, the coupled sECOM–MDO model accurately captures the temporal evolution of water elevation during Hurricane Sandy. Comparisons between observed and calculated storm surges at the Battery and Sandy Hook, shown in Fig. 9, also demonstrate that the performance of the model is favorable. The storm surge was estimated by subtracting the astronomical tide data at the NOAA gauges from the total water elevation.

Observed (red solid lines) and calculated (black dashed lines) time series of *η*_{t} (relative to NAVD88) induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (red solid lines) and calculated (black dashed lines) time series of *η*_{t} (relative to NAVD88) induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (red solid lines) and calculated (black dashed lines) time series of *η*_{t} (relative to NAVD88) induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (red solid lines) and calculated (black dashed lines) time series of storm surge *η*_{s} (relative to NAVD88) induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (red solid lines) and calculated (black dashed lines) time series of storm surge *η*_{s} (relative to NAVD88) induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Observed (red solid lines) and calculated (black dashed lines) time series of storm surge *η*_{s} (relative to NAVD88) induced by Hurricane Sandy.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Spatial variations of temporal maximum total water elevation and wave height are plotted in Fig. 10. While the maximum water elevation in the apex of the New York Bight is between 2 and 3 m, it is between 3 and 4 m in NY–NJ harbor. In contrast to water elevation that increases within the harbor, the wave height significantly reduces, mainly due to extensive wave breaking. The maximum wave height is between 6 and 11 m in the apex of the New York Bight, between 3 and 6 m near the mouth of NY–NJ harbor, and smaller than 3 m in most regions within the harbor. In Upper New York Bay, the maximum significant wave height is about 1.1 m. The maps of maximum water elevation and wave height, if produced before a predicted storm event, are very useful information for decision-making and evacuation planning for flood risk management.

(top) Maximum water elevation and (bottom) maximum *H*_{s} in NY–NJ harbor, calculated by the coupled sECOM–MDO model for Hurricane Sandy. Water elevation is relative to NAVD88.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

(top) Maximum water elevation and (bottom) maximum *H*_{s} in NY–NJ harbor, calculated by the coupled sECOM–MDO model for Hurricane Sandy. Water elevation is relative to NAVD88.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

(top) Maximum water elevation and (bottom) maximum *H*_{s} in NY–NJ harbor, calculated by the coupled sECOM–MDO model for Hurricane Sandy. Water elevation is relative to NAVD88.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

To evaluate the importance of wave-induced radiation stress and bottom friction included in the circulation model, we compare the results from the control run, which are shown in Figs. 8 and 9, with the results from a set of numerical experiments summarized in Table 2. While the control run considers the impacts of wave radiation stress and wave-enhanced bottom friction on currents, run 1 neglects wave radiation stress, run 2 neglects wave-enhanced bottom friction, and run 3 neglects both wave radiation stress and wave-enhanced bottom friction. In all experiments, the circulation and wave models remain coupled through the water depth and velocity field passed from sECOM to MDO and wave-enhanced water surface roughness computed in sECOM. Figure 11 compares the results from the control run with those from runs 1–3 at the Battery. The top panels reveal that the inclusion of wave radiation stress in the circulation model leads to an increase in the modeled water elevation, that is, wave setup, which is due to momentum transfer from breaking waves into the mean momentum. However, the contribution of wave setup to the total water elevation is small compared to the contribution of storm surge. The middle panels show that run 2, which neglects wave-enhanced bottom friction, overestimates the storm tide and surge, especially during 30 October, when the peak surge occurs. The intense wave–current interactions in shallow regions of the continental shelf enhance the bottom friction felt by currents. Neglecting the wave-enhanced bottom friction leads to lower bottom stress and turbulence, which allow, in turn, more volume of water to be transported to the harbor and, consequently, higher water elevation. The bottom panels in Fig. 11 also show that neglecting both wave radiation stress and wave-enhanced bottom friction in the circulation model causes considerable errors in the calculated storm tide and surge. Two-dimensional models usually neglect the contribution of waves to bottom stress, while the results presented here suggest this omission can be a weakness for creating physically accurate simulations of water levels in NY–NJ harbor during Hurricane Sandy.

Numerical experiments to assess impacts of waves on the circulation model.

Modeled (left) total water elevation and (right) surge from the control run (black dashed lines) and runs 1–3 (red solid lines) at the Battery. Control run: wave radiation stress and wave-induced bottom friction are included in sECOM; run 1: wave radiation stress is not included in sECOM; run 2: wave-induced bottom friction is not included in sECOM; run 3: wave radiation stress and wave-induced bottom friction are not included in sECOM.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Modeled (left) total water elevation and (right) surge from the control run (black dashed lines) and runs 1–3 (red solid lines) at the Battery. Control run: wave radiation stress and wave-induced bottom friction are included in sECOM; run 1: wave radiation stress is not included in sECOM; run 2: wave-induced bottom friction is not included in sECOM; run 3: wave radiation stress and wave-induced bottom friction are not included in sECOM.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Modeled (left) total water elevation and (right) surge from the control run (black dashed lines) and runs 1–3 (red solid lines) at the Battery. Control run: wave radiation stress and wave-induced bottom friction are included in sECOM; run 1: wave radiation stress is not included in sECOM; run 2: wave-induced bottom friction is not included in sECOM; run 3: wave radiation stress and wave-induced bottom friction are not included in sECOM.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

In addition to the comparisons described above, we quantitatively assess the influence of wave radiation stress and wave-enhanced bottom friction on the total water elevation. Table 3 summarizes the statistical metrics of the model performance at the Battery and Sandy Hook. Run 3, which neglects the effects of wave radiation stress and wave-enhanced bottom friction on the circulation model, shows the highest RMSEs. The inclusion of these effects in the circulation model decreases the RMSE from 0.17 (for run 3) to 0.12 m (for the control run) at the Battery and from 0.14 m to 0.08 m at Sandy Hook. Run 1, which neglects the wave radiation stress terms, shows the highest bias. Although not shown here, the statistical metrics calculated for the modeled storm surge are identical to the values shown in Table 3.

Statistical assessment of storm tides calculated by the coupled sECOM–MDO model (control run and runs 1–3) for Hurricane Sandy between 28 and 31 Oct 2012.

*η*

_{w}to

*η*by computing the difference between the total water elevations from the control run and the model without radiation stress gradient terms (run 1). Thus, the space- and time-dependent wave setup/setdown

*η*

_{w}(

*x*,

*y*,

*t*) is calculated as

*η*

_{w,max}(

*x*,

*y*). The temporal maximum wave setup/setdown in a particular computational cell represents the maximum amplitude of

*η*

_{w}(

*x*,

*y*,

*t*) over all time steps while the sign of

*η*

_{w}(

*x*,

*y*,

*t*) is preserved. In NY–NJ harbor, the maximum wave setup was between 0.2 and 0.3 m. In most regions in the apex of the New York Bight, the contribution of wave radiation stress to the total water elevation was to decrease it by 0.2 m. The model results indicate that the contribution of wave setup to the peak water elevation at the Battery and Sandy Hook were 0.13 and 0.10 m, respectively. The results also indicate that the maximum wave setup at these sites was 0.26 m (Fig. 13). The maximum wave setup at the Battery occurred at 0425 UTC 30 October, the time that the peak storm tide was receding the region and the water elevation and significant wave height at this site were 1.62 and 0.59 m, respectively. At Sandy Hook, the maximum wave setup occurred at 0345 UTC 30 October when the water elevation and significant wave height were 1.70 and 1.0 m, respectively. At the time that the maximum wave setup occurred at the Battery and Sandy Hook, the calculated wave heights offshore (buoy 44065) were 7.3 and 7.6 m, respectively. Thus, we may conclude that during Hurricane Sandy, the maximum wave setup at the Battery and Sandy Hook was about 3.5% of the maximum wave height at buoy 44065.

Model results of maximum wave setup/setdown (m) in NY–NJ harbor during Hurricane Sandy. Positive and negative values represent wave setup and setdown, respectively (see details in text).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Model results of maximum wave setup/setdown (m) in NY–NJ harbor during Hurricane Sandy. Positive and negative values represent wave setup and setdown, respectively (see details in text).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Model results of maximum wave setup/setdown (m) in NY–NJ harbor during Hurricane Sandy. Positive and negative values represent wave setup and setdown, respectively (see details in text).

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Calculated times series of *H*_{s} and *η*_{w} at the Battery and Sandy Hook.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Calculated times series of *H*_{s} and *η*_{w} at the Battery and Sandy Hook.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

Calculated times series of *H*_{s} and *η*_{w} at the Battery and Sandy Hook.

Citation: Journal of Atmospheric and Oceanic Technology 34, 7; 10.1175/JTECH-D-17-0005.1

The model results shown in Fig. 13 reveal approximately quarterdiurnal fluctuations in the wave setup time series at the Battery and Sandy Hook. The fluctuations can be due to quarterdiurnal processes, such as tidally driven bed stress or water column turbulent kinetic energy, that can be enhanced on both flood and ebb tides, as well as wave drag and wave radiation stress. Or, they could be caused by water-level-driven movement of the wave breaking zone relative to the deep channel and surrounding shoals (Fig. 6b). Further study would be required to determine the role of these processes.

## 4. Potential model applications and improvements

The successful coupling of MDO with sECOM is a promising step toward adopting the MDO wave model in flood hazard assessment studies and EFSs, such as the SFAS. Typically, EFSs should carry out hundreds of simulations every forecast cycle to have sufficient accuracy and lead time for actions to be taken to reduce flood risks. Therefore, they require computationally fast circulation and wave models. However, because of the high computational cost of full-spectrum wave models, the currently available EFSs are usually based on a stand-alone circulation model or, at best, a circulation model coupled with a simplified wave model. For example, the NOAA Probabilistic Hurricane Storm Surge model (http://www.nhc.noaa.gov/surge/psurge.php) and the Met Office Ensemble Prediction System (Flowerdew et al. 2010) use stand-alone depth-averaged circulation models and, therefore, neglect the impacts of surface waves on the total water elevation. The 3D circulation model used in SFAS, on the other hand, is coupled with the NOAA Great Lakes Environmental Research Laboratory (GLERL) wave model (Donelan 1977; Georgas et al. 2007). GLERL is a simple wave model based on wave momentum equations and is computationally fast. However, this model does not account of some important wave physical processes, such as swell propagation, wave radiation stress, and depth- and current-induced refraction.To accurately take into account the influence of waves on the predicted water levels, the circulation models used in EFSs must be properly coupled with a wave model that captures the most important wave physical processes but is still computationally inexpensive. The present study has shown that MDO can be successfully coupled with a circulation model and, therefore, may be considered for use in EFSs such as SFAS. A literature review shows that the third-generation wave models have been also successfully coupled with circulation models (e.g., Dietrich et al. 2011; Warner et al. 2008) and, therefore, may be considered as alternatives to MDO. However, these models take a higher computational cost compared to MDO. This is because a full-spectrum wave model solves the wave energy balance equation over the geographical (*x* and *y*), directional (*θ*), and frequency (*f*) spaces, whereas MDO solves the wave energy equation over geographical (*x* and *y*) and directional (*θ*) spaces and parameterizes the energy distribution over the *f* space. Therefore, for example, in the computational domain of NY–NJ harbor studied in the previous section (with 147 × 452 cells in the geographical space and 24 directional bins), while MDO solves the wave energy equation about 1.6 × 10^{6} times in every computational time step, a full-spectrum third-generation wave model with, for instance, 30 frequency bins would solve the wave energy equation about 4.8 × 10^{7} times.

To further investigate the speed of MDO compared to a full-spectrum third-generation wave model, we apply MDO and SWAN to an idealized case study of wind-generated waves and compare their computational costs. The idealized case is a 200 km ×200 km basin with constant water depth of 10 m. The computational domain is represented by a regular structured grid with a grid spacing of 1 km. We use 24 directional bins in MDO and SWAN, and 30 frequencies in SWAN. A time step of 30 s and two iterations per time step are used in both models. The models are run on a single CPU. The only driving force is due to a southerly wind with a constant speed of 15 m s^{−1}. Physical parameters/processes in SWAN are the default options (breaking, friction, whitecapping, and wave–wave interactions are turned on). The simulation period is 3 h. We found that while both models produce nearly similar results, SWAN is about 58 times more expensive than MDO. When we reduce the number of frequencies from 30 to 20, SWAN is about 41 times more expensive than MDO. These idealized experiments confirm the impact of the additional dimension (frequency space) on the computational cost of a phase-averaged wave model.

As described previously, MDO parameterizes the wave energy distribution in frequency space based on the DHHS spectrum, which contains elements of the JONSWAP spectrum. Similar to other empirical spectral models, JONSWAP and DHHS have their limitations. These wave spectra represent the sea state with a single-peak spectrum. Therefore, the MDO wave model should be used with caution in applications where a detailed spectral shape must be predicted, such as studies of long-term sediment transport and beach profile changes. If in such applications the sea state is characterized by a double-peaked spectrum, a phase-resolving wave model or a full-spectrum phase-averaged wave model should be used. However, even in the presence of a double-peaked spectrum, the significant wave height and mean wave period calculated by MDO may satisfactorily represent the sea surface statistics. In flood forecasting, where our interest is only in the impact of large waves for rare, localized extreme storms, it is expected that a single-peak spectrum will be sufficient.

The current version of the MDO wave model uses explicit numerical methods to discretize and solve the governing equations. Therefore, the computational time step used in MDO must satisfy the Courant–Friedrichs–Lewy (CFL) stability condition. The model results for the simulation of storm tides and waves generated by Hurricane Sandy, presented in the previous section, are based on sECOM–MDO with a coupling interval of every internal mode time step—that is, 10 s—for which the time step in MDO easily satisfies the stability condition. As a result, coupling of the wave model slowed our hydrodynamic model runs by a factor of 6. We carried out an additional simulation with a coupling interval of 30 s (results are not shown in this paper) and found that while the model results remain stable and accurate, MDO slows the hydrodynamic model by only a factor of about 3. Because of the small grid spacing used in the computational domain and, consequently, the violation of CFL stability condition, a larger coupling interval and time step in MDO (e.g., 50 s) causes numerical instability issues. Moving forward, it can be useful to improve the numerical methods used in MDO so that a larger coupling interval in sECOM–MDO can be adopted, further speeding the model execution.

The computational cost of sECOM–MDO can be also improved by using a parallel MPI version of the model. The source code of the current version of sECOM–MDO is in its serial version and is compiled using PGI’s OpenMP autoparallel Fortran compiler on 8 Intel(R) Xeon(R) CPUs (2.93 GHz). A parallel MPI version of sECOM–MDO can significantly reduce the computational cost. Jordi et al. (2017) have recently developed the MPI version of sECOM, which is currently being used in SFAS. A MPI-sECOM run on 8 CPUs is about 2.5 times faster than OpenMP autoparallel sECOM (for the same NYHOPS grid). The same research team is planning to develop the MPI version of MDO for use in operational flood forecasting.

## 5. Summary and conclusions

We have coupled sECOM with the MDO wave model for flood modeling in situations where storm tides and waves coexist. sECOM is the 3D circulation model used in the New York Harbor Observing and Prediction System (NYHOPS). Its parallel version (Jordi et al. 2017) is used in the Stevens Flood Advisory System (SFAS), an ensemble flood forecasting system that predicts meteorological and oceanographic conditions in New York City area and coastal New Jersey. The MDO wave model is a phase-averaged wave model developed by Mellor et al. (2008) with the intention of coupling it to 3D ocean circulation models. This wave model is simplified compared to popular third-generation wave models, and it was shown previously to be accurate (Mellor et al. 2008). The parameterization of wave energy distribution in frequency space makes MDO computationally cost effective and thus suitable for use in ensemble forecasting systems.

sECOM and MDO are coupled at the source code level and share a similar computational grid (Arakawa C grid with sigma vertical coordinate). The two models influence each other through water elevation, velocity field, wave radiation stress, and bottom and wind stresses. The feedback of sECOM to MDO consists of the input of water depth and velocity field. MDO responds to sECOM by returning the wave properties and depth-dependent radiation stresses. sECOM uses the wave properties to estimate the water surface roughness and the wave-enhanced bottom stress.

We have evaluated the coupled sECOM–MDO model using existing laboratory data. Good agreements between measurements and model results were achieved, which reveal the model’s capability to simulate wave characteristics, wave-induced water elevation, and undertow current. The model has been also evaluated using storm tides, surges, and waves generated by Hurricane Sandy in NY–NJ harbor. Qualitative and quantitative comparisons showed that sECOM–MDO accurately simulates hurricane waves and storm tides. The model results revealed that the temporal maximum wave setup in NY–NJ harbor was 0.26 m and that the contribution of wave setup to the peak water level was 0.13 m. It was found that the maximum wave setup and peak water level did not coincide. The statistical assessment of the model performance showed that the inclusion of wave radiation stress and wave-enhanced bottom friction reduced the RMSEs of the calculated storm tides from 0.17 to 0.12 m at the Battery and from 0.14 to 0.08 m at Sandy Hook.

To improve the accuracy of flood prediction and optimally mitigate flood risks under a changing climate, the next generation of EFSs should properly take into account the influence of waves—for example, wave setup—on the total water elevation. The satisfactory performance demonstrated here for the coupled sECOM–MDO model indicates that the MDO wave model can be successfully coupled with 3D circulation models. The simplicity and cost effectiveness of MDO, while it accurately captures the most important wave physical processes, can make it a suitable choice for use in EFSs and probabilistic flood hazard assessments.

## Acknowledgments

Work by RM and PO was funded by the U.S. Department of the Interior and the National Parks Service (Cooperative Agreement P14AC01472). Work by NG and AB was funded by a research task agreement entered between the Trustees of the Stevens Institute of Technology and the Port Authority of New York and New Jersey, effective August 19, 2014. We thank Oceanweather Inc. for providing the meteorological reanalysis data for Hurricane Sandy. We also would like to acknowledge the High Performance Computing Center of the City University of New York (CUNY), which is operated by the College of Staten Island and funded, in part, by grants from the city of New York, the state of New York, the CUNY Research Foundation, and the National Science Foundation Grants CNS-0958379, CNS-0855217, and ACI 1126113.

## APPENDIX

### Wave–Current Bottom Friction Coefficient

The effective wave-current bottom friction coefficient *C*_{cw} is determined through the following iterative steps:

- Step 1. Compute the wave-only bottom stress as
, where *ρ*is the water density,*U*_{*,w}is the wave friction velocity,*C*_{w}is the wave friction factor, and*U*_{w,b}is the wave orbital speed calculated based on the linear wave theory. Term*C*_{w}is calculated using the semiempirical relation of Signell et al. (1990) aswhere*A*_{δ}=*U*_{w,b}/*f*is the excursion amplitude,*f*is the wave frequency,*k*_{s}= 30*z*_{0}is the physical bottom roughness, and*z*_{0}is the bottom roughness length. Step 2. Compute the current-only friction factor

*C*_{c}(no wave effects) as*C*_{c}= 2[*κ*/ln(30*δ*_{cw}/*k*_{s})]^{2}, where*κ*= 0.41 is the von Kármán’s constant and*δ*_{cw}is the reference height of the wave–current bottom boundary layer.Step 3. Calculate the friction velocity for currents as

*U*_{*c}= (0.5*C*_{c})^{0.5}*U*_{b}, where*U*_{b}is the near-bottom mean current.Step 4. Calculate the combined current–wave friction velocity as

*U*_{*cw}= (*U*_{*c}^{2}+*U*_{*w}^{2})^{0.5}.Step 5. Calculate the apparent bottom roughness

*k*_{b}, which indicates the turbulence level due to the combination of wave–current boundary layer, and the physical bottom roughness as*k*_{b}=*k*_{s}[24(*U*_{*cw}/*U*_{w,b})(*A*_{δ}/*k*_{s})]^{β}, where*β*= 1 −*U*_{*c}/*U*_{*cw}.Step 6. Calculate

*C*_{cw}= 2[*κ*/ln(30*δ*_{cw}/*k*_{b})]^{2}.Step 7. If the difference between the most recent calculated

*C*_{cw}and*C*_{c}is smaller than a predefined threshold (e.g., 10^{−6}), then go to step 8, otherwise set*C*_{c}=*C*_{cw}and go back to step 3.Step 8. Store the most recent updated

*C*_{cw}and stop the iterative process.

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