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

Georgios Deskos; Joseph C. Y. Lee; Caroline Draxl; Michael A. Sprague

doi:10.1175/JAS-D-21-0003.1

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. The marine atmospheric boundary layer structure and governing equations
3. Wave-phase-averaged, wall-modeled LES
- a. Wave-age and wave-steepness dependence of the sea surface roughness
- b. Wave-spectra-informed sea surface
4. Wave-phase-resolved LES
5. Measurement data for model validation
6. Discussion
- a. The role of high-fidelity modeling in advancing our understanding of wind–wave interaction
- b. Knowledge gaps and limitations of MABL LES
7. Concluding remarks and future direction
Acknowledgments
APPENDIX
- Numerical Implementation of the HOS Method and Further Details on the Wave Field Initialization and Generation
  - a. High-order spectral solver
  - b. Wave field initialization
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Fig. 1.

Schematic representation of the atmospheric vertical structure over ocean waves.

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Fig. 2.

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

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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 T_p = 10 s, assuming that the atmospheric pressure fluctuations forcing term is zero $(p_{atm}^{'} / ρ_{a} = 0)$ . The computational domain has dimensions of 20λ_p × 20λ_p 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).

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Fig. 4.

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

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Fig. 5.

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

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Fig. 6.

Comparison of wave growth rates according to Miles’s theory with high-fidelity DNS and LES numerical simulations.

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Fig. 7.

Snapshot of the velocity and vorticity atmospheric fields over a swell. The numerical simulation was obtained using the open-source software framework Nalu-Wind (Sprague et al. 2020).

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Fig. 8.

Variation of the vertical-flux ratio −(Q₂ + Q₄)/(Q₁ + Q₃) (defined via quadrant analysis) with wave age c_p/(U_r cosϕ). The figure is adopted from Sullivan et al. (2008).

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Editorial Type:: Article

Article Type:: Research Article

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

Georgios Deskos

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

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https://orcid.org/0000-0001-7592-7191

Joseph C. Y. Lee

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

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Caroline Draxl

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

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, and

Michael A. Sprague

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

Received:: 05 Jan 2021

Accepted:: 14 May 2021

Published Online:: 08 Sep 2021

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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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.

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

Abstract

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),

z_{0} = \frac{a_{ch} u_{*}^{2}}{g},

(1)

where g is the gravitational acceleration, u_* is the friction velocity at the air–sea interface, and a_ch is the Charnock parameter. According to the IEC-61400-3 standard, the designer is given a choice between a_ch = 0.011 for open sea and a_ch = 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.

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:

\frac{\partial {\tilde{u}}_{i}}{\partial x_{i}} = 0,

(2a)

\frac{\partial {\tilde{u}}_{i}}{\partial t} + \frac{\partial}{\partial x_{j}} ({\tilde{u}}_{i} {\tilde{u}}_{j}) + 2 ε_{i j k} Ω_{j} ({\tilde{u}}_{k} - G_{k}) = - \frac{1}{ρ_{0}} \frac{\partial {\tilde{p}}^{'}}{\partial x_{j}} δ_{i j} - \frac{\partial τ_{i j}^{D}}{\partial x_{j}} + (\frac{\tilde{θ} - θ_{0}}{θ_{0}}) g_{i}, and

(2b)

\frac{\partial \tilde{θ}}{\partial t} + \frac{\partial}{\partial x_{j}} ({\tilde{u}}_{j} \tilde{θ}) = - \frac{\partial}{\partial x_{j}} q_{j},

(2c)

where the tilde represents a resolved-scale fluid property; u_i is the velocity field; ρ₀ is a reference density; p′ is the pressure fluctuation; Ω_j is Earth’s rotation vector; G_i is the geostrophic wind vector;

τ_{i j}^{D} = \tilde{u_{i} u_{j}} - {\tilde{u}}_{i} {\tilde{u}}_{j}

is the deviatoric part of the subgrid-scale stress tensor; q_j is the subgrid-scale heat flux; θ and θ₀ are the potential temperature and its reference value, respectively; and g_i = (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

τ_{i j}^{D}

, 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

τ_{tot} = ρ_{a} C_{d} | U_{r} | U_{r},

(3)

where C_d is a drag coefficient, ρ_a is the air density, and U_r 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 τ_turb, the wave-induced shear stress τ_wave, and the viscous shear stress τ_visc, so that the sum of all three is equal to the square of the friction velocity

u_{*}^{2}

τ_{tot} (z) = τ_{turb} (z) + τ_{wave} (z) + τ_{visc} (z) = ρ_{a} u_{*}^{2} .

(4)

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

τ_{turb} (z) = ρ_{a} {(κ z)}^{2} \frac{d u}{d z} | \frac{d u}{d z} |,

(5)

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

G_{d} (z)

of the wall (z = 0) wave-induced stress

τ_{wave} (z) = τ_{wave} (0) G_{d} (z),

(6)

which can, in turn, be defined through a directional wave spectrum F(k, ϕ), for example,

τ_{wave} (0) = ρ_{a} \int_{0}^{\infty} \int_{- π}^{π} ω^{2} F (k, ϕ) β (k, ϕ) d k d ϕ,

(7)

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 τ_wave(z) decays very rapidly with height z, and Janssen (1989) showed that the wind profile can be defined as

u (z) = \frac{u_{*}}{κ} \ln (\frac{z + z_{0} - z_{b}}{z_{0}}),

(8)

where z_b is the “background” roughness. Note that z_b 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 = 0.1ν_air/u_* and the “sea surface” roughness z₀,_w so that

z_{0} = z_{0, s} + z_{0, w} .

(9)

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),

τ_{i 3}^{wall} = - C_{d} U_{wall} {\tilde{u}}_{i}, for i = 1, 2,

(10)

where

U_{wall} = \sqrt{{\tilde{u}}_{1}^{2} + {\tilde{u}}_{2}^{2}}

is the magnitude of the wall-parallel velocity and C_d is a drag coefficient defined as

C_{d} = {[\frac{κ}{\log (z_{r} / z_{0}) - Ψ_{M} (z_{0} / L)}]}^{2},

(11)

where z_r is a reference height above the sea surface, Ψ_M is the Monin–Obukhov momentum similarity function, and

L = - u_{*}^{3} θ_{0} κ / (g Q_{0})

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

q_{θ}^{wall} = C_{h} U_{wall} (θ_{0} - \tilde{θ}),

(12)

where

C_{h} = \frac{u_{*} κ}{\log (z_{r} / z_{0}) - Ψ_{H}}

(13)

is the enthalpy coefficient and Ψ_H is the Monin–Obukhov enthalpy similarity function to correct for stability effects. To estimate z₀ (hereinafter we shall assume that z₀ ≈ z_0,w because z_0,s ≪ z_0,w), Charnock (1955) proposed a constant α_ch and related the roughness length scale as

z_{0} = \frac{α_{ch}}{g} u_{*}^{2} .

(14)

The values of α_ch 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 c_p/u_*, where c_p 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

z_{0}^{*} = A_{1} {(u_{*} / c_{p})}^{B_{1}},

(15)

where

z_{0}^{*} = z_{0} u_{*}^{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,

\frac{z_{0}}{H_{s}} = 3.35 {(\frac{u_{*}}{c_{p}})}^{3.4},

(16)

where H_s is the characteristic wave height. An alternative scaling to that of the inverse wave age u_*/c_p was also proposed by Taylor and Yelland (2001), who based it on wave steepness H_s/λ_p using the peak wavelength scale λ_p. It takes the form

\frac{z_{0}}{H_{s}} = A_{2} {(\frac{H_{s}}{λ_{p}})}^{B_{2}} .

(17)

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 c_p/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

ω_{p}^{*} \equiv ω_{p} u_{*} / g

, the nondimensional angular peak frequency. This is obtained from the inverse wave age (u_*/c_p) using the linear dispersion relationship of deep-water waves

ω_{p}^{2} = k g

Table 1.

Table of models for the sea surface roughness length scale.

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

z_{0}^{2} = A^{2} \int_{0}^{\infty} F (k) \exp [- 2 κ \frac{c (k)}{u_{*}}] d k,

(18)

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 = (k_x, k_y) 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

F_{J} (k) = \frac{a_{J}}{2 k^{3}} \exp [- \frac{5}{4} {(\frac{k_{p}}{k})}^{2}] γ^{r},

(19)

where k_p is the wavenumber of the spectrum peak, and a_J, 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) F_{j} (k),

(20)

where ϕ is the wave-propagation angle. Kitaigorodskii (1973) replaced the integrated form with

z_{0}^{*} = 0.068 {(u_{*} / c_{p})}^{- 3 / 2} \exp (- κ c_{p} / u_{*}),

(21)

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 (u_o, υ_o, w_o), 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, $p_{atm}^{'} = p_{atm} - \bar{p_{atm}}$ , 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

\nabla^{2} Φ = 0.

(22)

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:

\frac{\partial Φ}{\partial t} = - g h - \frac{1}{2} {| Φ |}^{2} - \frac{p_{atm}^{'}}{ρ_{w}} at z = h (x, y, t) and

(23a)

\frac{\partial h}{\partial t} = - \frac{\partial Φ}{\partial z} - \nabla Φ \cdot \nabla h at z = h (x, y, t) .

(23b)

Here, h(x, y, t) represents the free-surface amplitude measured from a reference height, z = 0, ρ_w is the water density, and

p_{atm}^{'}

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:

\frac{\partial h}{\partial t} = (1 + {| \nabla_{⊥} h |}^{2}) W - \nabla_{⊥} Φ_{s}^{2} \cdot \nabla_{⊥} h and

(24a)

\frac{\partial Φ_{s}}{\partial t} = - g h - \frac{1}{2} {| Φ_{s} |}^{2} + \frac{1}{2} (1 + {| \nabla_{⊥} h |}^{2}) W^{2} - \frac{p_{atm}^{'}}{ρ_{w}},

(24b)

where ∇_⊥ is the horizontal gradient operator and Φ_s = 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,

\frac{\partial Φ}{\partial z} (x, y, z = - d, t) = 0,

(25)

while periodicity is considered for the velocity potential Φ, W, and h along the two lateral directions:

[h, Φ_{s}, W] (

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GENERAL CONDITIONS

No. III

BAROMETRIC PRESSURE

VERIFICATIONS

MISCELLANEOUS PHENOMENA

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

Abstract

Abstract

1. Introduction

2. The marine atmospheric boundary layer structure and governing equations

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

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

b. Wave-spectra-informed sea surface

4. Wave-phase-resolved LES

a. Potential-flow theory and wave models