Editorial Type:
Review Article
Article Type:
Review Article

Review of Wind–Wave Coupling Models for Large-Eddy Simulation of the Marine Atmospheric Boundary Layer

Georgios Deskos aNational Wind Technology Center, National Renewable Energy Laboratory, Golden, Colorado

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Joseph C. Y. Lee aNational Wind Technology Center, National Renewable Energy Laboratory, Golden, Colorado

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Caroline Draxl aNational Wind Technology Center, National Renewable Energy Laboratory, Golden, Colorado
bRenewable and Sustainable Energy Institute, Boulder, Colorado

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Michael A. Sprague aNational Wind Technology Center, National Renewable Energy Laboratory, Golden, Colorado

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Online Publication:
08 Sep 2021
Print Publication:
01 Oct 2021
DOI:
https://doi.org/10.1175/JAS-D-21-0003.1
Page(s):
3025–3045
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Abstract

We present a review of existing wind–wave coupling models and parameterizations used for large-eddy simulation of the marine atmospheric boundary layer. The models are classified into two main categories: (i) the wave-phase-averaged, sea surface–roughness models and (ii) the wave-phase-resolved models. Both categories are discussed from their implementation, validity, and computational efficiency viewpoints, with emphasis given on their applicability in offshore wind energy problems. In addition to the various models discussed, a review of laboratory-scale and field-measurement databases is presented thereafter. The majority of the presented data have been gathered over many decades of studying air–sea interaction phenomena, with the most recent ones compiled to reflect an offshore wind energy perspective. Both provide valuable data for model validation. We also discuss the modeling knowledge gaps and computational challenges ahead.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Georgios Deskos, georgios.deskos@nrel.gov

Abstract

We present a review of existing wind–wave coupling models and parameterizations used for large-eddy simulation of the marine atmospheric boundary layer. The models are classified into two main categories: (i) the wave-phase-averaged, sea surface–roughness models and (ii) the wave-phase-resolved models. Both categories are discussed from their implementation, validity, and computational efficiency viewpoints, with emphasis given on their applicability in offshore wind energy problems. In addition to the various models discussed, a review of laboratory-scale and field-measurement databases is presented thereafter. The majority of the presented data have been gathered over many decades of studying air–sea interaction phenomena, with the most recent ones compiled to reflect an offshore wind energy perspective. Both provide valuable data for model validation. We also discuss the modeling knowledge gaps and computational challenges ahead.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Georgios Deskos, georgios.deskos@nrel.gov
Keywords: Wind waves; Marine boundary layer; Large-eddy simulations; Atmosphere–ocean interaction; Parameterization; Renewable energy

1. Introduction

Wind–wave interaction has been a topic of continuous research for almost a century: from Jeffreys’s sheltering hypothesis (Jeffreys 1925) to the most recent high-fidelity, wave-phase-resolved, and turbulence-resolving numerical simulations (Hao and Shen 2019) and large-scale field campaigns (Black et al. 2007; Edson et al. 2007). As many physical processes rely on accurately quantifying the heat, mass, and momentum exchanges that take place at the air–sea interface, wind–wave interaction dynamics have been cardinal to many scientific and engineering disciplines: from climate and weather prediction to plankton generation models (Sullivan and McWilliams 2010; Cavaleri et al. 2012). A relatively new field that is expected to also be affected by wind–wave interaction is offshore wind energy. Currently, the total installed offshore wind energy capacity of the United States amounts to 30 MW and is expected to increase drastically over the next decade, with up to 14 GW of capacity planned to be installed off the U.S. East Coast by 2030 (Musial et al. 2019). The installation and operation of offshore large-scale wind farms have raised numerous questions about the predictability of wind and wave resources in offshore environments, including our ability to predict the mean wind speed and turbulence levels that future wind turbines will experience as well as the wave field relevant to floating offshore wind turbine dynamics. To this end, wind–wave interaction lies in the epicenter of offshore wind energy. First, wind–wave interaction together with other prevalent marine boundary layer phenomena (e.g., low-level jets) can draw a completely different picture for the dynamics of atmospheric turbulence than its land-based counterpart. For instance, a recent study by Bodini et al. (2019) showed that the turbulence levels experienced by offshore wind turbines in the U.S. East Coast can be considerably lower than those found onshore. Second, the combined aerodynamic and hydrodynamic loading on floating offshore wind turbines introduces the need to also simulate both effects simultaneously for either operational (Butterfield et al. 2007) or extreme, nonoperational event scenarios (Kim et al. 2016). In any case, better models for wind–wave interaction are key to reducing uncertainty in the simulation of offshore wind energy systems and therefore further contribute to their financial viability (Veers et al. 2019).

Despite its crucial and omnipresent role in the offshore environment, wind–wave interaction has not been fully explored for the purposes of wind turbine design. As recent as 2009, the International Electrotechnical Commission (IEC) established an offshore wind turbine international standard (IEC 2009a,b), which is used for the design of both fixed-bottom and floating wind turbines (IEC-61400-3-1, IEC-61400-3-2). Within these guidelines, the designer has to specify parameters such as a reference wind speed and turbulence intensity at hub height, averaged over a period of 10 min, and refer back to IEC (2009a) (a land-based wind turbine standard) for estimating the wind speed profile. Wind–wave interaction enters the standards only in the form of a “sea surface” roughness parameter z0, which is calculated using the well-known Charnock relation (Charnock 1955),
z0=achu*2g,
where g is the gravitational acceleration, u* is the friction velocity at the air–sea interface, and ach is the Charnock parameter. According to the IEC-61400-3 standard, the designer is given a choice between ach = 0.011 for open sea and ach = 0.034 for near-coastal waters. Even more interesting, most high-fidelity simulations of offshore wind turbines have also ignored the role of the coupled wind–wave dynamics (Churchfield et al. 2012; Nilsson et al. 2015; Wu and Porté-Agel 2015; Deskos et al. 2020). In fact, almost all of the state-of-the-art offshore wind farm simulators are missing the capability of resolving the phase of the underlying traveling waves (Breton et al. 2017) and only a handful of studies have considered the effect of wind–wave interaction on the power and loads of offshore wind turbines (Yang et al. 2014a,b; AlSam et al. 2015; Calderer et al. 2018).

In recent years, wind–wave interaction research has enjoyed the renewed interest from both the engineering and scientific communities. This is evident by the increasing number of large-scale field campaigns (Black et al. 2007; Edson et al. 2007) and the development of novel numerical models capable of resolving and incorporating wave motions in atmospheric turbulence simulations (Yang and Shen 2011a,b; Sullivan et al. 2014). Wave-resolving simulations are relevant to turbulence-resolving atmospheric modeling via large-eddy simulation (LES) of the airflow over regular and irregular waves. It should be noted here that large-eddy simulations of the planetary boundary layer have been performed for almost five decades and are known to yield high-fidelity data for canonical cases (e.g., flow over a horizontally homogeneous flat terrain) including the daytime buoyancy-driven boundary layer (Deardorff 1972; Moeng 1984; Mason 1988; Moeng and Sullivan 1994), the neutrally stratified atmospheric boundary layer (ABL) (Mason and Thomson 1987; Andren et al. 1994; Pedersen et al. 2014), and the nocturnal stable boundary layer (Kosović and Curry 2000; Mason and Derbyshire 1990; Basu and Porté-Agel 2006). The high accuracy of LES is because of to the fact that the larger flow structures are sufficiently resolved at the computational grid scale while smaller subgrid-scale structures can be accurately modeled (Meneveau and Katz 2000). LES has also been successfully employed to study atmospheric turbulence over complex terrain (Fedorovich 1986; Smith and Skyllingstad 2005; Chow et al. 2006). More recently, however, wave-phase-resolved simulations of the marine atmospheric boundary layer (MABL) have been undertaken (Sullivan et al. 2008, 2014, 2018b; Hao et al. 2018; Hao and Shen 2019), which have enhanced our understanding of the offshore turbulence characteristics. Wave-phase-resolved simulations require the use of more sophisticated numerical algorithms often comprising two separate but loosely coupled subsolvers, one to predict the temporal/spatial evolution of the free-surface waves and another to simulate the atmospheric turbulence or alternatively a two-phase (i.e., air/water) solver with an advanced interface-tracking algorithm (Prosperetti and Tryggvason 2009). The latter approach has not been used in large-eddy simulation of the MABL and only a handful of direct-numerical simulation (DNS) studies exist to this day (Lin et al. 2008; Yang et al. 2018). However, the former approach that employs the two subsolvers loosely coupled together at the air–sea interface is considered to be a state-of-the-art model for large-eddy simulations of the MABL. Its use in marine boundary layer simulations has helped us gain insight into the complex wind–wave interaction processes and to that extent it has also challenged the validity of the widely used Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954) in wave-driven wind cases (Sullivan and McWilliams 2010; Jiang et al. 2016). We note here that MOST provides an analytical expression for the mean velocity and temperature profiles in the lower part of the ABL also known as the “surface layer” (see section 2) and therefore is germane to wind energy applications. Considering the importance of wind–wave interactions in predicting the ambient atmospheric turbulence as well as the impact they may have on predicting turbine loads and power output, as it was recently demonstrated by Yang et al. (2014a,b), AlSam et al. (2015), and Calderer et al. (2018), we argue that wind–wave coupling models remain a key aspect of offshore wind energy modeling and therefore need to be thoroughly studied.

In this article, we present a comprehensive review of wind–wave coupling models that can be adopted to perform large-eddy simulations of the MABL within the context of turbulence-resolving simulations at grid scales of magnitude O(1–15 m) and used in offshore wind energy applications at similar or finer resolution. We shall refer to such scale-resolving simulations within domain sizes of O(1–50 km3) as “microscale” simulations. This article complements many existing reviews on the fluid dynamics of wind over waves and the marine boundary layer (Belcher and Hunt 1998; Sullivan and McWilliams 2010) and textbooks and proceedings on wind–wave interaction (Csanady and Gibson 2001; Jones and Toba 2001; Hunt and Sajjadi 2003; Janssen 2004). However, in this review, we emphasize the modeling aspects by laying out all of the crucial algorithmic components. Thereby, the remainder of this review article starts with section 2, where we introduce the main geophysical processes present in the MABL together with the governing equations used within a three-dimensional, unsteady LES framework. Next, we present the wave-phase-averaged and phase-resolved modeling techniques in sections 3 and 4, respectively. These are followed in section 5 by a brief introduction of the most recent field databases collected in offshore regions of high relevance to offshore wind energy. In section 6, we summarize the physical insights gained from performing large-eddy simulations of the MABL as well as the remaining knowledge gaps and modeling limitations of the reviewed models.

2. The marine atmospheric boundary layer structure and governing equations

The MABL is a special type of the ABL that occurs over oceans or large lakes. As such, it shares many of its features with the well-studied, land-based ABL, including a similar vertical structure. On the other hand, the physical processes of the MABL are more complex as they involve additional processes at the air–sea interface that span multiple time and length scales. Nonetheless, for the sake of simplicity, we define the vertical structure of the MABL in analogy with the land-based ABL (Stull 1988; Kaimal and Finnigan 1994; Wyngaard 2010) by splitting it into two distinct regions: the “surface” or “constant-flux” layer and a free-atmosphere-topped “mixed” layer, as shown in Fig. 1. The former is about 1–100 m high, and the vertical variations of the turbulent fluxes therein are not expected to exceed 10% of their surface values. This region can be further split into the “wave boundary layer,” which is the region immediately adjacent to, and directly impacted by, the motion of the waves, and the “inertial sublayer,” where the bulk exchange of mass, momentum, heat, and moisture between the wind and waves takes place. The wave boundary layer is considered to be only a few meters high [O(1 m)] for wind-driven waves and it is believed to reach higher altitudes only in cases when light winds blow over faster-moving swells (Grachev and Fairall 2001; Grachev et al. 2003). The inertial sublayer, together with the wave boundary layer, make up approximately 10%–20% of the overall boundary thickness. Above the surface layer lies the mixed layer, which, depending on the thermal stratification, can extend to altitudes of 200–700 m. Last, the free atmosphere, a stably stratified nonturbulent region above the atmospheric boundary layer, lies atop the mixed layer and its onset defines the overall MABL thickness. For completeness, we should also mention that a thin layer above the air–sea interface exists, called the surface or viscous sublayer, where viscous fluid forces are important. Thanks to the very-high-Reynolds-number nature of the MABL, Re > 107, this region is only a few millimeters thick and therefore ignored in most numerical models.

Fig. 1.
Fig. 1.

Schematic representation of the atmospheric vertical structure over ocean waves.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0003.1

The dynamics of the MABL can be described by the unsteady, compressible Navier–Stokes equations, which is common in mesoscale models [e.g., Weather Research and Forecasting (WRF) Model; Skamarock et al. 2019]. However, in microscale modeling, one can assume acoustic incompressibility to define a potential temperature θ that remains constant during isentropic displacements in the atmosphere and employ the Boussinesq approximation to account for buoyancy effects in the momentum equation. The resulting simplified model provides a framework for large-eddy simulation of the MABL, and it may take the following form:
∂u˜i∂xi=0,
∂u˜i∂t+∂∂xj⁡(u˜iu˜j)+2εijkΩj⁡(u˜k−Gk)=−1ρ0∂p˜′∂xjδij−∂τijD∂xj+(θ˜−θ0θ0)gi, and
∂θ˜∂t+∂∂xj⁡(u˜jθ˜)=−∂∂xj qj,
where the tilde represents a resolved-scale fluid property; ui is the velocity field; ρ0 is a reference density; p′ is the pressure fluctuation; Ωj is Earth’s rotation vector; Gi is the geostrophic wind vector; τijD=uiuj˜−u˜iu˜j is the deviatoric part of the subgrid-scale stress tensor; qj is the subgrid-scale heat flux; θ and θ0 are the potential temperature and its reference value, respectively; and gi = (0, 0, −g) is the gravitational acceleration vector. It is worth noting that in the LES equations of the ABL, the viscous term has been ignored because of the high-Reynolds-number nature of the flow. Subgrid-scale stress closure models have been proposed for τijD, from the standard Smagorinsky model (Smagorinsky 1963) to the one-equation kinetic energy model (Moeng 1984), to more sophisticated dynamic models (Meneveau and Katz 2000). The effect of the moving waves can be introduced into the previously mentioned model using two different approaches. The first is that of the wave-phase-averaged and wall-modeled LES, where a flat-bottom computational domain is used, and the wave motions are averaged in time, whereas the second and more computationally demanding approach is to resolve the wave phase and represent it in the airflow solver through a deforming “waving” boundary. The latter approach will be called the wave-phase-resolved LES model. The numerical and modeling approaches are discussed in the following sections (3 and 4).

3. Wave-phase-averaged, wall-modeled LES

In wave-phase-averaged MABL, an analogy between momentum transfers in the vicinity of a solid rough surface and that of the air–sea interface can be made. Thus, for aerodynamic purposes, moving waves can be described through roughness elements. For both microscale and mesoscale modeling of the atmospheric boundary layer, surface fluxes are often represented by the bulk aerodynamic drag so that
τtot=ρaCd|Ur|Ur,
where Cd is a drag coefficient, ρa is the air density, and Ur is a reference velocity above the sea surface. Following the analysis of Phillips et al. (1966) and Makin and Kudryavtsev (1999), the total stress can be further decomposed into the turbulent shear stress τturb, the wave-induced shear stress τwave, and the viscous shear stress τvisc, so that the sum of all three is equal to the square of the friction velocity u*2,
τtot⁡(z)=τturb⁡(z)+τwave⁡(z)+τvisc⁡(z)=ρau*2.
With these three terms supporting wind shear, the viscous shear stress is often neglected because of the high-Reynolds-number characterizing the ABL. The turbulent stress is often parameterized through the mixing-length theory, which suggests that
τturb⁡(z)=ρa⁡(κz)2dudz|dudz|,
where κ ≈ 0.41 is the von Kármán constant and du/dz is the velocity gradient at height z. The wave-induced stress, on the other hand, may be defined through a decaying function Gd⁡(z) of the wall (z = 0) wave-induced stress
τwave⁡(z)=τwave⁡(0)Gd⁡(z),
which can, in turn, be defined through a directional wave spectrum F(k, ϕ), for example,
τwave⁡(0)=ρa∫0∞∫−ππω2F⁡(k,ϕ)β⁡(k,ϕ) dk dϕ,
where k and ϕ are the wavenumber and propagation angle of the waves, respectively, and β(k, ϕ) is the wave growth function. For wind-driven waves, it can be argued that τwave(z) decays very rapidly with height z, and Janssen (1989) showed that the wind profile can be defined as
u⁡(z)=u*κ ln⁡(z+z0−zbz0),
where zb is the “background” roughness. Note that zb becomes z0 when the wave-induced stress becomes very small. This is the case in most microscale studies (Smith 1988; Fairall et al. 2003), in which the roughness length scale, is defined as the sum of the viscous-supported length scale z0,s = 0.1νair/u* and the “sea surface” roughness z0,w so that
z0=z0,s+z0,w.
This roughness scale can be used in wall-modeled LES to represent the near-surface fluid dynamics through a wall-stress model (Schumann 1975; Moeng 1984),
τi3wall=−CdUwallu˜i,fori=1,2,
where Uwall=u˜12+u˜22 is the magnitude of the wall-parallel velocity and Cd is a drag coefficient defined as
Cd=[κlog⁡(zr/z0)−ΨM⁡(z0/L)]2,
where zr is a reference height above the sea surface, ΨM is the Monin–Obukhov momentum similarity function, and L=−u*3θ0κ/⁡(gQ0) the Obukhov length scale defined through the friction velocity u* and the heat flux Q0 at the air–sea interface. In a similar fashion, we may define the heat flux at the air–sea interface through a surface heat flux
qθwall=ChUwall⁡(θ0−θ˜),
where
Ch=u*κlog⁡(zr/z0)−ΨH
is the enthalpy coefficient and ΨH is the Monin–Obukhov enthalpy similarity function to correct for stability effects. To estimate z0 (hereinafter we shall assume that z0 ≈ z0,w because z0,s ≪ z0,w), Charnock (1955) proposed a constant αch and related the roughness length scale as
z0=αchg u*2.
The values of αch have been reported to vary from 0.0144 to 0.035 (Kitaigorodskii and Volkov 1965; Garratt 1977; Wu 1980; Geernaert et al. 1986) with these differences being attributed to a number of factors such as the wind speed (Fairall et al. 2003; Edson et al. 2013), the sea state (e.g., wave age) (Donelan 1990; Smith et al. 1992; Fairall et al. 2003; Oost et al. 2002), or the water depth (Geernaert et al. 1986, 1987; Geernaert 1990; Smith et al. 1992; DeCosmo et al. 1996; Taylor and Yelland 2001; Foreman and Emeis 2010; Jiménez and Dudhia 2018). Waves tend to be rougher over shallow waters thanks to the open ocean.

a. Wave-age and wave-steepness dependence of the sea surface roughness

Many studies have already tried to relate z0 to the peak (or characteristic) wave age cp/u*, where cp is the peak wave-phase speed (Masuda and Kusaba 1987; Donelan 1990; Toba et al. 1990; Smith et al. 1992; Drennan et al. 2005), and all have proposed a relationship that takes the form
z0*=A1⁡(u*/cp)B1,
where z0*=z0u*2/g is a friction-velocity-scaled roughness and A1 and B1 are the two experimentally determined model parameters. These parameters have been found to vary significantly from dataset to dataset and there is no universally accepted set of values (for more details see Table 1). This is related to the fact that in Eq. (15) both the wave roughness z0 and the wave-phase velocity are scaled by the friction velocity u*, as was noticed by Smith et al. (1992) and Lange et al. (2004). As a remedy to this problem, Drennan et al. (2003) grouped data by u* so that any variability in wave age would arise from the waves peak phase velocity, and they proposed the following relation,
z0Hs=3.35⁡(u*cp)3.4,
where Hs is the characteristic wave height. An alternative scaling to that of the inverse wave age u*/cp was also proposed by Taylor and Yelland (2001), who based it on wave steepness Hs/λp using the peak wavelength scale λp. It takes the form
z0Hs=A2⁡(Hsλp)B2.
The same authors found that the parameter values A2 = 1200 and B2 = 3.4 fit many existing datasets so that this new formulation does not exhibit a spurious correlation between z0 and u*. However, it was also found that their model performs poorly for very short fetches (young waves with cp/u* < 15). Alternative formulations of the wave-steepness scaling exist and often utilize the root-mean-square of the surface elevation, instead (Donelan 1990). A number of the above-mentioned models are presented in Table 1 using both the original Charnock (1955) or the significant wave height, scaling for normalizing the sea surface roughness. In addition, we plot in Fig. 2 all of the models presented in Table 1 using both scalings, as well as existing field data from the literature. We note here that the nondimensional sea surface roughness is plotted against ωp*≡ωpu*/g, the nondimensional angular peak frequency. This is obtained from the inverse wave age (u*/cp) using the linear dispersion relationship of deep-water waves ωp2=kg.
Table 1.

Table of models for the sea surface roughness length scale.

Table 1.
Fig. 2.
Fig. 2.

Compilation of models for sea surface roughness as a function of the nondimensional angular peak frequency.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0003.1

b. Wave-spectra-informed sea surface

A more systematic way to compute sea surface roughness was presented by Kitaigorodskii (1968), who used the wavenumber spectrum F(k) to obtain
z02=A2∫0∞F⁡(k) exp[−2κc⁡(k)u*] dk,
where c(k) is the wave-phase speed of wavelets with a wavenumber, k, and A is a model constant/parameter. The significance of Kitaigorodskii’s model is that the integration of the roughness components (wavelets) are filtered by their wave age through the exponential term exp[−2κc(k)c(k)/u*], also known as the Kitaigorodskii filter. Hence, during wave growth and the shift of the wave spectral peak to lower wavenumbers the contribution to the overall roughness length is transferred to the spectral tail. Thus, to obtain the roughness length scale z0 from Eq. (18), an empirical wave-spectrum model needs to be used. This may include a unidirectional-wave spectrum or directional-wave variance spectrum, F(k), where k = (kx, ky) is the wavenumber spectrum. Such a spectrum is that of the Joint North Sea Wave Project (JONSWAP), which in its one-dimensional form in wavenumber space is given by
FJ⁡(k)=aJ2k3 exp[−54⁡(kpk)2]γr,
where kp is the wavenumber of the spectrum peak, and aJ, γ, and r are parameters of the sea state. The final wave spectrum formulation takes into account the directionality of the waves by multiplying the one-dimensional spectrum with a spreading function D(k, ϕ),
S⁡(k,ϕ)=D⁡(k,ϕ)Fj⁡(k),
where ϕ is the wave-propagation angle. Kitaigorodskii (1973) replaced the integrated form with
z0*=0.068⁡(u*/cp)−3/2 exp⁡(−κcp/u*),
which, however, is not consistent with the integration of Eq. (18). Other variances of the integral form can be sought through directly integrating the wave spectra, and examples can be found in the studies of Donelan et al. (1985), Banner (1990), and Elfouhaily et al. (1997).

4. Wave-phase-resolved LES

The sea surface–roughness approach provides a static representation of the wind–wave interaction and can be interpreted as a wave-informed drag model that only accounts for a downward momentum transfer (from wind to waves). However, recent studies have shown that this is not the case under swell-dominated seas where long waves propagate into light or moderate winds and where momentum can also be transferred upward (Grachev and Fairall 2001; Sullivan et al. 2008). In such a scenario, a dynamic coupling between the airflow and the propagating waves is needed. Such approaches have been pursued for airflow over monochromatic or broadband waves in either the context of direct-numerical simulations (Sullivan et al. 2000; Sullivan and McWilliams 2002; Shen et al. 2003; Kihara et al. 2007; Yang and Shen 2009, 2010; Druzhinin et al. 2012; Åkervik and Vartdal 2019; Wang et al. 2020), wall-resolved large-eddy simulations (Zhang et al. 2019; Cao et al. 2020), or wall-modeled LES (Sullivan et al. 2008; Yang et al. 2013a; Sullivan et al. 2014; Hara and Sullivan 2015; Sullivan et al. 2018a,b; Hao et al. 2018; Hao and Shen 2019). The latter approach is appropriate for the dynamic coupling between wind and ocean waves in the context of the MABL and requires the use of two separate models: a wall-model-equipped LES solver to resolve the atmospheric turbulent flow and a potential-flow solver to advance the free-surface. The dynamic coupling between the two models is achieved by an exchange of information at the air–sea interface. In particular, the potential-flow wave model provides information about the free-surface vertical displacement η(x, y, t) and local orbital velocity (uo, υo, wo), which in turn enters the LES solver through a waving boundary condition. The updated sea surface elevation distorts the LES mesh by moving the mesh nodes vertically to remap the computational domain. On the other hand, the LES model passes the surface atmospheric pressure fluctuations, patm′=patm−patm¯, to the potential-flow wave solver, which is used as a dynamic boundary condition on the free surface. This approach is often referred to as “two-way coupling.” In the following sections we present, for both models, their governing equations as well as modeling approaches and the coupling procedure necessary to yield the final wind–wave interaction model.

a. Potential-flow theory and wave models

To describe the evolution and propagation of water waves in the open ocean, it is often desirable to assume that the flow be incompressible, inviscid, and irrotational, which admits potential-flow theory. To this end, we may describe the wave velocity field U = ∇Φ through a potential velocity field Φ(x, y, z, t) so that
∇2Φ=0.
Equation (22) is derived directly by applying a conservation of mass and assuming potential flow. Furthermore, if we assume that the wave free surface is a material surface, we may obtain the dynamic and kinematic free-surface boundary conditions, respectively:
∂Φ∂t=−gh−12|Φ|2−patm′ρwatz=h⁡(x,y,t)and
∂h∂t=−∂Φ∂z−∇Φ⋅∇hatz=h⁡(x,y,t).
Here, h(x, y, t) represents the free-surface amplitude measured from a reference height, z = 0, ρw is the water density, and patm′ is the atmospheric pressure fluctuations at the free surface. Solutions to Eq. (22) under the general form of the boundary conditions (23a) and (23b) cannot be obtained analytically because of the nonlinear nature of the dynamic free-surface boundary condition (DFSBC). Simpler solutions, however, are possible (Airy 1845; Stokes 1847; Fenton 1985) by linearization of the DFSBC and are often available in water-waves mechanics or marine hydrodynamics textbooks (e.g., Newman 1977; Dean and Dalrymple 1991). Alternative solutions include the use of numerical algorithms such as the finite-element method (Ma and Yan 2006), the boundary-element method (Longuet-Higgins and Cokelet 1976), or the high-order spectral (HOS) method (Dommermuth and Yue 1987). The latter, although it is restricted to periodic boundary conditions, is often the preferred wave model for wave-phase-resolved MABL simulations thanks to its higher accuracy and efficiency. HOS algorithms utilize the Zakharov equations (Zakharov 1968) to describe the deep-water nonlinear free-surface boundary conditions:
∂h∂t=(1+|∇⊥h|2)W−∇⊥Φs2⋅∇⊥hand
∂Φs∂t=−gh−12|Φs|2+12⁡ (1+|∇⊥h|2)W2−patm′ρw,
where ∇⊥ is the horizontal gradient operator and Φs = Φ[x, y, h(x, y, t), t] and W = (∂Φ/∂z)(x y, h, t) are the free-surface velocity potential and vertical velocity, respectively. Additionally, in HOS models the vertical potential gradient needs to vanish at a water depth z = −d,
∂Φ∂z⁡ (x,y,z=−d,t)=0,
while periodicity is considered for the velocity potential Φ, W, and h along the two lateral directions:
[h,Φs,W]⁡(x=0,y,t)=[h,Φs,W]⁡(x=Lx,y,t)and
[h,Φs,W]⁡(x,y=0,t)=[h,Φs,W]⁡(x,y=Ly,t).
The numerical implementation of the HOS method as well as further details on the wave field initialization and generation can be found in the appendix sections a and b, respectively. An example of a solution obtained via the open-source code HOS-ocean for the purpose of this review is shown in Fig. 3.
Fig. 3.
Fig. 3.

Example of an irregular wave field produced by an HOS model. The presented results correspond to irregular JONSWAP waves, with a peak period Tp = 10 s, assuming that the atmospheric pressure fluctuations forcing term is zero ⁡(patm′/ρa=0). The computational domain has dimensions of 20λp × 20λp and is discretized using 256 × 256 nodes. The presented snapshot was originally generated for the purposes of this review using the open-source code HOS-ocean (Ducrozet et al. 2016).

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0003.1

b. Computational fluid dynamics solvers using mesh motion

The need to resolve the flow within the MABL introduces the need for a waving boundary condition, i.e., an otherwise solid boundary is moving either in a prescribed fashion or in response to the airflow pressure field. Such a condition requires the numerical solvers to be able to handle mesh motion through either finite-volume/element frameworks that are based on arbitrary Lagrangian–Eulerian methods (Noh 1963; Donéa et al. 1982) or through grid-coordinate transformations (Anderson 1995, chapter 5). The latter approach has been the preferred option in the wind–wave interaction literature and a number of different algorithms have been developed (Yang and Shen 2011a,b; Sullivan et al. 2014) as a straightforward extension of existing pseudospectral or finite-difference-based computational fluid dynamics (CFD) solvers. Here, we present the governing equations, coordinate transformation, and overall implementation by adopting the notation and numerical algorithms from Sullivan et al. (2014). A similar problem formulation can be found in the work of Yang and Shen (2011a,b). Nonetheless, to obtain a wave-following system of equations we need to apply a coordinate transformation between the “physical-space” coordinate system (x, y, z, t) and the computational space coordinate system (ξ, η, ζ, t) [xi=(x,y,z,t)⇔ξi=(ξ,η,ζ,t)] in which
x=ξ,y=η,z=ζ+h⁡(x,y,t)⁡(1−ζLz)υand
∂ξ∂x=1,∂ζ∂x=−∂z∂ξ,∂ζ∂z=J,and∂z∂t=−∂ζ∂t1J,
where h(x, y, t) is the time-varying surface wave height, and here υ = 3 is the mesh dampening coefficient (Fig. 4).
Fig. 4.
Fig. 4.

Schematic representation of the coordinate transformation between the physical and computational domains. The figure is adopted from Yang et al. (2013a).

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0003.1

With these definitions, we may rewrite the LES Eqs. (2a), (2b), and (2c) as
∂Ui∂ξi=0,
∂∂t⁡(u˜iJ)+∂∂ξj [⁡(Uj−δ3jzt)u˜i]=FiJ,and
∂∂t⁡(θ˜J)+∂∂ξj[⁡(Uj−δ3jzt)θ˜]=QiJ,
where Fi and Qi are right-hand-side momentum and heat terms defined via
FiJ=−∂∂ξj⁡(p˜′J∂ξj∂xi)−∂∂ξj⁡(τijDJ∂ξj∂xi)+(θ˜−θ0Jθ0)gi+2εijkΩj ⁡(u˜k−Gk)J gand
QiJ=−∂∂ξj⁡(τiθJ∂ξj∂xi).
Here, both the momentum and potential-temperature advection terms are written in the strong flux-conservation form using the contravariant flux velocity Ui = (U, V, W),
Ui=ujJ∂ξi∂xj.
The incompressibility condition is guaranteed by also solving for the pressure Poisson equation,
∂∂ξi⁡(1J∂ξi∂xj∂ξm∂xj∂p*∂ξm)=S,
where S is a force proportional to the divergence field of the advection term.

1) Geometric conservation law

Another important aspect of the “wave following” solver is the satisfaction of the so-called geometric conservation law (GCL), which should guarantee that the discrete system of equations under mesh motion conserves its volume (Thomas and Lombard 1979; Demirdžić and Perić 1990). Thus, in a coordinate transformation formulation, the GCL needs to satisfy the following condition:
∂∂t⁡(1J)+∂∂ξi⁡(1J∂ξi∂t)=0.
Furthermore, using the previously described coordinates transformation, the GCL further simplifies to
∂∂t⁡(1J)−∂∂ζ⁡(∂z∂t)=0.
General-purpose CFD algorithms that do not utilize the “coordinate transformation” approach will need to follow a more generic formulation (e.g., Mavriplis and Yang 2006).

2) Boundary conditions and surface fluxes

While the shape of the wavy surface and its movement partly define the boundary conditions through mesh motion, the velocity and wall-stress (surface flux) boundary conditions also need to be considered. It is worth emphasizing here, that although the wave-phase-resolved simulations resolve the wave-coherent structures, they also rely on wall modeling to account for the subgrid-scale wave motions. Starting with the velocity boundary condition, the total time rate of change of the wave height h(x, y, t) is
DzDt|fs=DhDt=wo=∂h∂t+∂h∂x uo+∂h∂y υoatζ=0.
If we require an impenetrable boundary condition in the wall-normal direction (ζ direction), we then need to set the contravariant velocity, W = ∂h/∂t. Similarly, the top velocity boundary condition is set to W = zt = 0 at ζ = ZL. Because no mesh motion takes place under the other two directions (streamwise and lateral), the two velocity boundary conditions are simply set to U = uo and V = υo. Next, stresses/surface fluxes at the wall can be computed using the wall-stress models described in section 3. These surface fluxes, however, need to be computed in the local wave-fitted coordinate system assuming the law-of-the-wall formulas. Thus, at each grid node, we define the two tangential vectors, t1 and t2, a surface normal vector, n = t1 × t2, as well as a relative filtered velocity vector, u˜s=u˜w−u˜a as the difference between the water waves velocity u˜w and the airflow velocity u˜a above it. To that end, the shear-stress tensor is defined as
τ′=−Cd|u˜s|[u˜s⁡(1)t1+u˜s⁡(2)t2],
where the prime represents a shear-stress tensor aligned with the local, transformed coordinate system and the surface heat as
Q*=Ch|u˜s|⁡(θ0−θ˜).
The bulk-transfer coefficients, Cd and Ch, are again computed using the MOST arguments of a flat surface. One parameter that needs to be entered here is z0. Only this time z0 does not represent the roughness coming from all wavelength scales but only of the unresolved ones. Most studies with wave-phase-resolved simulations have adopted small values—for example, z0 = 2 × 10−4 in Sullivan et al. (2008)—to represent the roughness coming from the unresolved waves. However, Yang et al. (2013a,b) investigated five different dynamic sea surface roughness models, which take into account the underlying wave spectra, namely, the “root-mean-square model,” the “geometry model,” the “steepness-dependent Charnock model,” the “wave-kinematics-dependent model,” and the “combined-kinematics-steepness model.” They found that the wave-kinematics-dependent model based on the idea of Kitaigorodskii and Volkov (1965), which accounts for both the wave amplitude information (through the wave spectrum function) and the kinematics of wind and wave relative motion (through an exponential function that depends on the ratio of wave-phase speed to wind-friction velocity), yields the best performance.

c. Wave motion via the immersed boundary method

An alternative solution to the mesh-motion wind–wave coupling can be sought by using the immersed boundary method (IBM), first introduced by Peskin (1972). Since its conception, the IBM has been used in numerous fluid mechanics problems and engineering applications (Mittal and Iaccarino 2005; Griffith and Patankar 2020) and it is in general considered a robust and mature approach to resolving the fluid flow over complex geometries. From an implementation point of view, IBM does not require geometry-conforming meshes and the influence of a solid boundary on the fluid flow is realized through a boundary force. In incompressible solvers, the force is applied during an intermediate time step and through the Poisson equation solved within the projection method to ensure that the flow field remains divergence-free everywhere in the domain. In the context of the atmospheric boundary layer, IBM has been used to resolve the airflow over complex terrain (Bao et al. 2018; Arthur et al. 2020; Liu and Stevens 2020) and in general two variants of the method have been proposed. In the first approach, the velocity just above the terrain surface is reconstructed to fit a profile given by similarity theory (e.g., MOST) and it is referred to as the velocity-reconstruction immersed boundary method (VR-IBM) (Bao et al. 2018; DeLeon et al. 2018), whereas in the second the shear stress in the vicinity of the immersed surface is reconstructed using similarity theory, and it is therefore referred to as the shear-stress reconstruction method (SR-IBM) (Diebold et al. 2013; Ma and Liu 2017; Liu and Stevens 2020). The latter approach resembles the boundary condition of the geometry-conforming moving meshes just mentioned earlier as surface fluxes are calculated along the immersed boundary using the local tangential and normal vectors. Finally, it is worth noting that while to the best of our knowledge the IBM has not been applied to wind–wave interaction problems yet, its extension to moving boundaries can be straightforward as previous studies on moving objects and fluid–structure interaction have already proved its feasibility (Griffith and Patankar 2020).

d. Interface between the wind LES and wave HOS models

Regardless of the airflow discretization strategy (geometry-conforming moving mesh/moving immersed boundary) a coupling procedure is needed to provide a two-way (or dynamic) coupling between the wind and wave models. Thus, at each time step, the wave field is forced by the wind by incorporating the atmospheric pressure fluctuations into the dynamic boundary condition. The wave solution is integrated in time for Δt, and the wave elevation h and orbital velocity field uo = (uo, υo, wo) are updated. These, in turn, are used by the LES model to adapt the geometry of the domain boundary (and its mesh thereafter), as well as apply a new velocity boundary condition at each mesh node. The LES model advances in time as well, and the coupling procedure is repeated. The interface is shown schematically in Fig. 5. Note here that a number of LES studies (e.g., Yang et al. 2013a) do not consider the airflow’s feedback to the wave model. This is because of the fact that the time scale for the wave to evolve under wind-driven conditions is much larger than the advection and turnover time scales of turbulent eddies and thus near-surface pressure variations (Liu et al. 2010). In that case, the magnitude of the time step needed by the two models may differ by an order of magnitude, with the LES solver often imposing a stricter Courant–Friedrichs–Lewy (CFL) number condition. For instance, in the very recent two-way-coupled study of Hao and Shen (2019) undertaken for the evolution of wind-driven (slow) waves, the time step of the coupled HOS-LES simulations was chosen to be 8.7 × 10−3 × Tp, where Tp is the waves’ peak period, whereas at the same time, the high-order spectral method of Dommermuth and Yue (1987) could have allowed for a larger time step to be used based on the local CFL number. On the other hand, in swell-dominant (fast moving) wave scenarios, the wave solver may be the one to dictate the time step size. To this end, from a numerical point of view, the two models may exchange information less or more frequently and oftentimes allow one of the solvers to advance its solution multiple time steps before exchanging information with the other one. Last, but not least, extra care should be taken to account for any differences in the spatial resolution used by the two models. In the simple case where the horizontal mesh resolution of the LES model is identical to the wave model, the mapping between the two models’ mesh is straightforward. However, oftentimes the two models may employ different degrees of refinement, thus, an interpolation algorithm may also be needed.

Fig. 5.
Fig. 5.

Schematic representation of the two-way coupling between the LES wind model and the potential-flow wave model.

Citation: Journal of the Atmospheric Sciences 78, 10; 10.1175/JAS-D-21-0003.1

5. Measurement data for model validation

Validation datasets have historically been critical to model development with field campaigns and their reported data extending over five decades: Atlantic Trade Wind Experiment (ATEX; Dunckel et al. 1974), Air Mass Transformation Experiment (AMTEX;