1. Introduction
Ocean surface waves play a key role in the dynamics of the marine atmospheric boundary layer (ABL), impacting the winds and scalar concentrations within the ABL by modulating the momentum exchange and scalar fluxes at the air–sea interface (D’Asaro and McNeil 2007). Understanding the interaction between ocean surface waves and the ABL has important ramifications for both fundamental science and engineering, including parameterization of wave impacts in weather and climate modeling (Large et al. 1994), marine weather predictions and offshore wind energy assessment (Yang et al. 2014a,b; Xiao and Yang 2019).
The effect of waves on the wind field depends on various wave properties, including wave steepness (characterized by ka, where k is the wavenumber and a is the wave amplitude), wave age (defined as c/U_{0} or
Wind–wave interaction has been studied with both one and twoway coupled numerical models. In oneway coupling, the wave field is imposed as a bottom boundary condition affecting the airflow, yet the waves are not influenced by the wind. Different types of waves have been employed in oneway coupling studies, including a monochromatic linear sinusoidal wave (e.g., Sullivan et al. 2000; Zhang et al. 2019; Cao and Shen 2021), Stokes waves (e.g., Yang and Shen 2011a; Druzhinin et al. 2019; Cao et al. 2023), and broadband waves (e.g., Sullivan et al. 2014, 2018a). In oneway coupling between broadband waves and airflow, researchers further neglect nonlinear wave–wave interactions and superimpose broadband linear wave components to accelerate computational speed (Sullivan et al. 2014, 2018b). Both direct numerical simulation (DNS) and largeeddy simulation (LES) have been employed in oneway coupled models, using a curvilinear coordinate that maps the grid to the shape of the moving waves (e.g., Sullivan et al. 2000, 2014; Zhang et al. 2019). These studies focus on the waveinduced effects on the mean wind profile, shear stress, and turbulence characteristics. In twoway coupled models, the wind and the wave are coupled dynamically and evolve together. Existing twoway coupled models use either a curvilinear coordinate (e.g., Yang and Shen 2011a,b; Li and Shen 2022b) or the volumeoffluid method (e.g., Campbell et al. 2016; Cimarelli et al. 2023; Wu and Deike 2021) to resolve the twophase flow. The majority of the twoway coupling studies employ DNS and focus on the smallscale dynamics of nonlinear wave–wave interaction, turbulence on the waterside, scalar transfer across the interface or wind–wave generation (Lin et al. 2008; Liu et al. 2009; Komori et al. 2010; Campbell et al. 2016; Wu and Deike 2021; Li and Shen 2022a,b). Although twoway coupled models resolve the influence of the wind on waves, they are computationally expensive and limited to smallscale dynamics at relatively low Reynolds numbers. They require complicated boundary conditions on the deformable water surface to enforce continuous velocity and stress profiles, and result in high computational costs (Yang and Shen 2011b; Campbell et al. 2016). Hao and Shen (2019) built an LES tool to simulate twoway coupled wind and wave fields. In their model, waves are simulated with a highorder spectral (HOS) method developed by Dommermuth and Yue (1987). This method captures both the evolution of phaseresolved waves under the influence of the wind and the nonlinear interaction between waves. Hao and Shen (2019) showed that the nonlinear interactions, rather than the influence of the wind, play the dominant role in the longterm wave evolution. The HOS wave solver needs to have a finer grid and smaller time step than the LES model because nonlinear wave–wave interaction transfers energy to higher frequencies. This increases the computational cost of the twoway coupled LES–HOS model, and the grid size mismatch between the LES and HOS models further complicates the implementation of the coupling. Moreover, Yang and Shen (2009) compared one and twoway coupled DNS simulation results and found negligible differences in airflow vortical structures between the two. These results suggest that the oneway coupled model can provide satisfactory fidelity much more efficiently than a twoway coupled model for the study of wave effects on airflow and turbulence.
Studies using either one or twoway coupled models have revealed important waveinduced effects on the ABL. Sullivan et al. (2000) showed that monochromatic waves can increase the turbulent momentum flux near the wave surface by as much as 40% compared to a flat surface. Using the LES–HOS model initialized with a wave field generated by an empirical ocean spectrum, Hao and Shen (2019) found that the streamwise velocity spectrum of the wind displays a clear wave signature that follows the dispersion relation for deep water waves, but this signature is restricted to a height dictated by the peak wavelength. The oneway coupled model has been used to show that ocean swells can increase the overall wind power extraction in offshore wind farms by as much as 18% in low wind speed conditions when the waves act to increase the wind speed (Yang et al. 2014a,b). Hao et al. (2018) utilized the same coupled model to explore the interactions between airflow and wave groups and found that the presence of long waves can reduce the form drag of short waves. This model has also been applied for simulating wind over fastpropagating monochromatic waves, revealing that the vertical component of the wave orbital velocity dominates the waveinduced airflow perturbation (Cao and Shen 2021). More recently, several oneway coupled numerical studies have focused on windopposing wave or misaligned wind and waves. Cao et al. (2023) used LES with prescribed Stokes waves and theoretical analysis to show that for fast opposing waves, waveinduced airflow perturbation is dominated by the linear response of the wind to the wave. Deskos et al. (2022) used DNS with linear, monochromatic sine waves of different phase speeds and directions relative to the wind. They reported a large deviation in the mean velocity vector relative to the applied pressure gradient when fastmoving waves were aligned at an angle of 135° relative to the applied pressure gradient vector. However, this DNS study is limited to a very low Reynolds number, similar to other twoway coupled DNS studies focusing on smallscale features such as the waveinduced Stokes sublayer (Cimarelli et al. 2023) and wind–wave growth mechanisms (Li and Shen 2022a,b; Wu et al. 2022).
The studies mentioned above have explored many aspects of wind–wave interaction. However, the simulations are limited to idealized or synthetic atmospheric conditions lacking important meteorological factors such as temperature stratification, humidity, and radiation. To develop a comprehensive understanding of the interaction between ocean surface waves and the ABL, a coupled wind–wave model that can simulate realistic meteorological and wave conditions is needed. A good candidate for realistic wind simulations is the Weather Research and Forecasting (WRF) Model. WRF is a widely used mesoscale numerical weather prediction model for both atmospheric research and operational applications (Skamarock et al. 2008). WRF solves the compressible Navier–Stokes equations with Coriolis terms and includes transport equations for temperature, moisture, and tracers. Using a pressurebased curvilinear coordinate, WRF can simulate flow over complex terrain with moderate slopes and over multiple scales from turbulenceresolving to synoptic. At large scales, WRF can be forced by numerous meteorological datasets and provides multiple numerical solution options as well as physics parameterizations for land surface, planetary boundary layer, atmospheric and surface radiation, microphysics, and cumulus convection. At small scales, an LES capability is also available with different subgridscale turbulence options. WRF also provides a gridnesting mesh refinement option that enables LES domains to be placed within mesoscale boundingdomain simulations, permitting smallscale process studies to be conducted within realistic mesoscale flows forced by real data. Moreover, WRF’s twoway nesting capability provides a tool to fully couple different scales with both upsale and downscale information exchange at nested domain boundaries.
Given these capabilities, coupling WRF–LES with phaseresolved moving surface waves can enable comprehensive numerical studies of the influence of surface waves on the ABL in both idealized and realistic meteorological and oceanographic settings. These studies will benefit largerscale atmospherewave modeling by aiding the improvement of drag parameterization for wind over surface water waves. In fact, Yang et al. (2013) developed a dynamic model of sea surface roughness using a twoway coupled LES–HOS model. A recent study proposed a less computationally expensive sea surface based drag model for LES of wind over waves, which is only applicable to monochromatic sinusoidal waves with a known phase speed and wave steepness (Aiyer et al. 2023). The coupled wave and WRF–LES model can be used to study drag parameterization under more realistic scenarios and extend the functionality of existing drag parameterization. Another potential application of the coupled wave and WRF–LES model is to improve offshore wind energy prediction. Castorrini et al. (2023) recently coupled a mesoscale numerical weather prediction model with a localscale Reynoldsaveraged Navier–Stokes (RANS) model to investigate offshore wind turbine inflow. Since wave effects are parameterized in their model, a coupled wave and WRF–LES model could improve the effect of the waves on the wind. In this manner, a coupled wave and WRF–LES model will be able to provide more accurate multifidelity simulation results for offshore wind energy applications.
As the current WRF model’s surface elevation boundary condition does not permit timevariability, this paper presents an implementation of a moving bottom to represent propagating water waves. The method is validated with test cases including laminar and turbulent flow over both stationary and moving sine waves. The paper is organized as follows: section 2 describes in detail the vertical coordinate and governing equations in WRF, followed by the moving bottom implementation; section 3 presents results of two laminar flow test cases and turbulent flow test cases with three different wave ages; and finally, section 4 summarizes the results and discusses future work.
2. Methods
a. Governing equations in WRF
The grid in WRF, adapted from Skamarock et al. (2008), where (i, j, k) are the indices of grid points in the (ξ, η, ζ) directions.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The grid in WRF, adapted from Skamarock et al. (2008), where (i, j, k) are the indices of grid points in the (ξ, η, ζ) directions.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The grid in WRF, adapted from Skamarock et al. (2008), where (i, j, k) are the indices of grid points in the (ξ, η, ζ) directions.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
For time advancement, WRF uses the Runge–Kutta (RK) method. Each RK step is divided into a number of acoustic steps that update the acoustic equations. The number of acoustic steps can be defined by the user based on the CFL condition related to advection and propagation of sound waves, as discussed in section 2d.
b. Moving bottom implementation
The implementation of a moving bottom in WRF is divided into two parts. The first is to specify boundary conditions, and the second is to update the hydrostatic balance at each time step as the bottom moves.
1) Bottom boundary conditions
2) Hydrostatic rebalancing
In the original WRF code, the hydrostatic variables
c. Numerical procedure
The numerical procedure for the moving bottom implementation in WRF is summarized here, and is slightly modified from Skamarock et al. (2008). In each RK step, the forcing terms F_{U}, F_{V}, and F_{W} and the

Advance horizontal momentum with Eqs. (16a) and (16b), with boundary conditions (28a) and (28b) for a rough surface and free or noslip, otherwise.

Advance μ″ with Eq. (16d) and compute Ω^{τ}^{+Δ}^{τ} after assuming Ω″ = 0 at the bottom and top boundaries.

Advance ϕ″ and W″ with Eqs. (16e) and (16c) and with boundary conditions (24) and (25).
After the acoustic update, p′ and α′ are updated based on Eqs. (11) and (14f) to make sure they satisfy the equation of state and the hydrostatic relation at every RK step. Finally, after all RK steps, we conduct hydrostatic rebalancing, and then the model advances to the next time step.
d. Model stability
3. Model validation
In this section, we present three test cases to validate the movingwave implementation in WRF–LES: 1) inviscid laminar flow, 2) viscous laminar flow and 3) turbulent flow. For all three cases, the bottom boundary is prescribed as a monochromatic linear water wave. The wave propagates in the streamwise direction, here taken to be the x direction, and there is no variation in the spanwise (y) direction. Cases 1 and 2 have analytical solutions of the airflow over moving waves, whereas case 3 does not have an analytical solution. Therefore, for case 3 we validate the code by comparing the movingwave simulation results to an equivalent case in a wavefollowing reference frame, such that it can be simulated with the original fixedbottom WRF–LES code that has already been extensively validated.
a. Inviscid flow over a linear water wave
As shown in Fig. 3, the x–z contours of u, w, and p′ show good agreement between the simulation results and the analytical solutions. The movingbottom WRF Model also reproduces the analytical vertical profiles of u, w, and p′ as shown in Fig. 4.
Validation of the inviscid linear sine wave: x–z contours of (a) u, (b) w, and (c) p′. The colored contours are WRF simulation results and the black contour lines are analytical solutions, taken at t/T = 17.67.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave: x–z contours of (a) u, (b) w, and (c) p′. The colored contours are WRF simulation results and the black contour lines are analytical solutions, taken at t/T = 17.67.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave: x–z contours of (a) u, (b) w, and (c) p′. The colored contours are WRF simulation results and the black contour lines are analytical solutions, taken at t/T = 17.67.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave: vertical profiles of (a) u, (b) w, and (c) p′. The red symbols are WRF simulation results and the black lines are analytical solutions. Different symbols refer to different time instances.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave: vertical profiles of (a) u, (b) w, and (c) p′. The red symbols are WRF simulation results and the black lines are analytical solutions. Different symbols refer to different time instances.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave: vertical profiles of (a) u, (b) w, and (c) p′. The red symbols are WRF simulation results and the black lines are analytical solutions. Different symbols refer to different time instances.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
As an additional validation of the inviscid flow case, we use the original WRF–LES code to simulate inviscid flow over a fixed wavy bottom, but initialize the horizontal velocity with u = −c everywhere so that it is equivalent to the movingwave case, except in a wavefollowing reference frame. We refer to this case as the movingframe case, and the previous case with a moving wave as the fixedframe case in the remainder of the paper. The movingframe case uses the same computational domain and resolution as the fixedframe case. We expect the results from the movingframe case to be the same as the fixedframe case after transforming the solutions to the same reference frame. To compare the results from the two cases, we assume that any transients in each solution have vanished, giving a steady solution for the movingframe case and a timeperiodic solution for the fixedframe case. Then we note that the streamwise velocity in the steady solution of the movingframe case, u_{m}(x, z), is related to the streamwise velocity for the fixedframe case, u_{f}(x, z, t), with u_{m}(x, z) + c = u_{f}(x − ct, z, t). Meanwhile, w_{m}(x, z) = w_{f}(x − ct, z, t) and
As shown in Fig. 5, the two cases agree with each other in terms of their vertical profiles of u, w, and p′, and also match the analytical solutions. The overshoot of w in the fixed wave case at the top of the domain is likely due to acoustic wave reflection because there is no damping applied at the top boundary. Figure 5 shows an example of the vertical profile comparison taken at x = 0 in the movingframe case, although the same level of agreement is attained for other locations as well.
Validation of the inviscid linear sine wave in different frames of reference: profiles of (a) u, (b) w, and (c) p′ of the fixedframe (blue circles) and the movingframe simulations (black plus signs), along with the analytical solution (red lines), taken at t/T = 20 and x = 0 in the fixed reference frame.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave in different frames of reference: profiles of (a) u, (b) w, and (c) p′ of the fixedframe (blue circles) and the movingframe simulations (black plus signs), along with the analytical solution (red lines), taken at t/T = 20 and x = 0 in the fixed reference frame.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Validation of the inviscid linear sine wave in different frames of reference: profiles of (a) u, (b) w, and (c) p′ of the fixedframe (blue circles) and the movingframe simulations (black plus signs), along with the analytical solution (red lines), taken at t/T = 20 and x = 0 in the fixed reference frame.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Having demonstrated that the method reproduces the linear inviscid theory, we now demonstrate the error convergence of the movingbottom WRF code using an inviscid wave with a = 0.16 m and λ = 100 m. The domain size is L = 100 m and H = 100 m. For temporal convergence, a spatial resolution of Δx = 2.5 m is used, and for spatial convergence, a temporal resolution of Δt = 0.0005 s is used. The simulations run for one second. Figures 6 and 7 show relative errors of u, w, and p′ as a function of Δt and Δx, respectively. The relative error is defined as the absolute difference in u, w, or p′ between each case and the case with the next smaller Δt or Δx. Given a location in x (at the centerline in y), we compute the relative error using three types of norms: the L_{1}, L_{2}, and L_{∞} norms computed with data over all grid points from ζ = 0 to 1. The convergence rates are sensitive to the error metric that is chosen, as shown in Figs. 6 and 7. The numerical values of the convergence rates are computed as the slopes of the lines in Figs. 6 and 7. They are listed in Table 1. Convergence rates indicated by the L_{1} and L_{∞} norms are almost the same for u and w, but different for p′. For temporal convergence, errors using the L_{2} norm indicate roughly secondorder accuracy for w, nearly thirdorder accuracy for u, and between first and secondorder accuracy for p′. The deviation from secondorder accuracy for u and p′ might be due to the fact that inviscid flow is sensitive to initial transients. For spatial convergence, errors using the L_{1} and L_{∞} norms indicate secondorder accuracy, while errors using the L_{2} norm indicate roughly fourthorder accuracy. Although we do not have a clear explanation for the behavior of the different norms, we can find error metrics that display secondorder accuracy for spatial convergence, consistent with the fact that WRF is secondorder accurate in space as discussed in section 2d.
Timestepping error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δt for the inviscid flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines indicate the following: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Timestepping error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δt for the inviscid flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines indicate the following: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Timestepping error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δt for the inviscid flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines indicate the following: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Spatial discretization error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δx for the inviscid flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line),secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Spatial discretization error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δx for the inviscid flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line),secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Spatial discretization error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δx for the inviscid flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line),secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Temporal and spatial convergence rates of the inviscid flow and the viscous laminar flow test cases for u, w, and p′ using L_{1}, L_{2}, and L_{∞} norms.
b. Viscous laminar flow over a linear wave
Figures 8 and 9 show good agreement between WRF simulation results and the analytical solutions for the viscous laminar flow test case. In Fig. 8a, strong shear is present in u in the boundary layer (z < δ) due to the Stokes boundary layer. Above the boundary layer, u and w transition to the inviscid solutions. The error in u between the WRF simulation results and the analytical solution has a magnitude of 0.05aω, which is 5% of the analytical solution (Fig. 9c). The error in w has a magnitude of up to only 0.01aω, which is 1% of the analytical solution (Fig. 9d). The errors are concentrated in the height below 5δ, above which they approach zero. Figure 9 shows vertical profiles of the viscous velocities u_{ν} and w_{ν} in the boundary layer, which again agree well with the analytical solutions. Because the simulation results and the analytical solution for p′ remain the same in the viscous and the inviscid flow cases, the results for p′ are not shown here.
The viscous linear sine wave case: x–z contours of (a) u, (b) w, (c) the error in u, and (d) the error in w between the WRF simulation results and the analytical solution. The colored contours are WRF simulation results and the black contour lines are analytical solutions, taken at t/T = 17.67. The y axis on the right shows the height normalized by the Stokes boundary layer thickness δ.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The viscous linear sine wave case: x–z contours of (a) u, (b) w, (c) the error in u, and (d) the error in w between the WRF simulation results and the analytical solution. The colored contours are WRF simulation results and the black contour lines are analytical solutions, taken at t/T = 17.67. The y axis on the right shows the height normalized by the Stokes boundary layer thickness δ.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The viscous linear sine wave case: x–z contours of (a) u, (b) w, (c) the error in u, and (d) the error in w between the WRF simulation results and the analytical solution. The colored contours are WRF simulation results and the black contour lines are analytical solutions, taken at t/T = 17.67. The y axis on the right shows the height normalized by the Stokes boundary layer thickness δ.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The viscous linear sine wave case: vertical profiles of (a) u_{ν} and (b) w_{ν}. The red symbols are WRF simulation results, and the black lines are analytical solutions. Different symbols refer to different times.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The viscous linear sine wave case: vertical profiles of (a) u_{ν} and (b) w_{ν}. The red symbols are WRF simulation results, and the black lines are analytical solutions. Different symbols refer to different times.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
The viscous linear sine wave case: vertical profiles of (a) u_{ν} and (b) w_{ν}. The red symbols are WRF simulation results, and the black lines are analytical solutions. Different symbols refer to different times.
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Using the same setup as the inviscid flow test case, discussed in section 3a, we show the temporal and spatial convergence of the viscous laminar flow case in Figs. 10 and 11. The numerical values of the convergence rates are computed as the slopes of the lines in Figs. 10 and 11. They are listed in Table 1. For temporal convergence (Fig. 10), errors using the L_{2} norm indicate roughly secondorder accuracy for w and p′, and thirdorder accuracy for u. The L_{1} and L_{∞} norms yield lower convergence rates that are between first and secondorder. For spatial convergence (Fig. 11), errors using the L_{∞} and L_{1} norms are parallel to each other for all variables and indicate secondorder accuracy, but errors using the L_{2} norm indicate fourthorder accuracy, similar to the inviscid case. Again, we do not have a clear explanation for the behaviors of the different norms, but at least one of the three error metrics show second or higherorder convergence for both time and space for the viscous laminar flow case. The spatial errors for the viscous laminar flow case form straighter lines (Fig. 11) than the spatial errors for the inviscid flow case (Fig. 7) and are slightly closer to the expected secondorder slope. This is likely because of the smoothing effect of viscosity, which reduces transients and gridscale noise that are more likely to be present in the inviscid test case.
Timestepping error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δt for the viscous laminar flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Timestepping error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δt for the viscous laminar flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Timestepping error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δt for the viscous laminar flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Spatial discretization error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δx for the viscous laminar flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Spatial discretization error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δx for the viscous laminar flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Spatial discretization error in (a) u, (b) w, and (c) p′ relative to the results from the case with the next smaller Δx for the viscous laminar flow test case. The errors are computed with three types of norms: L_{1} norm (black stars), L_{2} norm (black circles), and L_{∞} norm (black triangles). Lines are as follows: firstorder slope (red solid line), secondorder slope (red dashed line), and thirdorder slope (red solid line with dots).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
c. Turbulent flow over a monochromatic wave
In the turbulent flow test case, the wave height has the same form as Eq. (37), with a = 0.8 m and λ = 50 m, which results in a steepness of ka = 0.1. The computational domain is 300 m × 150 m × 100 m, with 120 × 60 × 80 grid points in the streamwise (x), spanwise (y) and vertical (z) directions. The horizontal grid spacing is Δx = Δy = 2.5 m. The vertical grid spacing is Δz = 0.5 m at the lowest level above the wave surface, and is continuously stretched by a factor of 1.02 from one grid cell to the next to the top of the domain. The time step size is Δt = 0.01 s, with n_{s} = 4. The Smagorinsky closure scheme is used to compute the turbulent eddyviscosity. The flow is driven by a constant pressure gradient forcing, which is set to achieve a desired wave age
The presence of surface waves has a notable effect on altering the turbulence structure of the airflow, as previously reported in numerical simulations (e.g., Sullivan et al. 2000; Yang and Shen 2009), as well as laboratory experiments (e.g., Cheung and Street 1988; Buckley and Veron 2016). Here, we present results of WRF simulations on the waveinduced turbulence quantities in the wind. Figure 12 shows contours of the waveinduced streamwise velocity
Contours of (left) normalized waveinduced streamwise velocity
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Contours of (left) normalized waveinduced streamwise velocity
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Contours of (left) normalized waveinduced streamwise velocity
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Figure 13 shows the stress profiles for the turbulent flow cases with the three different wave ages. The intermediate wave generates near zero wave stress and form drag, while decreasing the wave age leads to much stronger wave stress and form drag. The magnitudes of the wave stress and form drag are consistent with the small magnitude of
Stress profiles of turbulent flow over a moving sine wave with wave age (a)
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Stress profiles of turbulent flow over a moving sine wave with wave age (a)
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Stress profiles of turbulent flow over a moving sine wave with wave age (a)
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
To further validate our methodology, we compare the turbulent flow results with a moving wave in a fixed frame to a movingframe case with a fixed wavy bottom. Details of the fixed and movingframe cases are discussed in section 3a. For the movingframe case, the relative velocity u_{r} = u − c − u_{w} is used to transform the fixed frame to the moving frame, and then used to compute the bottom stress in Eqs. (28a) and (28b). The movingframe case has the same numerical setup as the fixedframe case. Figure 14 presents the results for
Normalized mean streamwise velocity and Reynolds stress profiles for the fixed and movingframe cases of turbulent flow with
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Normalized mean streamwise velocity and Reynolds stress profiles for the fixed and movingframe cases of turbulent flow with
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Normalized mean streamwise velocity and Reynolds stress profiles for the fixed and movingframe cases of turbulent flow with
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Wave growth rate at different wave ages. Red circles show WRF–LES results, black triangles show wallmodeled LES results from Hao and Shen (2019), plus signs are wallresolved LES from Cao and Shen (2021), black squares are DNS results from Kihara et al. (2007), black right triangles are Reynoldsaveraged Navier–Stokes (RANS) results from Li et al. (2000), and black circles are experimental data from Grare et al. (2013).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Wave growth rate at different wave ages. Red circles show WRF–LES results, black triangles show wallmodeled LES results from Hao and Shen (2019), plus signs are wallresolved LES from Cao and Shen (2021), black squares are DNS results from Kihara et al. (2007), black right triangles are Reynoldsaveraged Navier–Stokes (RANS) results from Li et al. (2000), and black circles are experimental data from Grare et al. (2013).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Wave growth rate at different wave ages. Red circles show WRF–LES results, black triangles show wallmodeled LES results from Hao and Shen (2019), plus signs are wallresolved LES from Cao and Shen (2021), black squares are DNS results from Kihara et al. (2007), black right triangles are Reynoldsaveraged Navier–Stokes (RANS) results from Li et al. (2000), and black circles are experimental data from Grare et al. (2013).
Citation: Monthly Weather Review 151, 11; 10.1175/MWRD230077.1
Although the explicit acoustic substepping algorithm in WRF requires a small model time step due to the fast sound speed, for our grid spacing we are able to use a time step of Δt = 0.01 s with n_{s} = 4, as discussed in section 2d. For the slow wave case with
4. Conclusions
In this study, we presented the implementation of a moving bottom in WRF–LES and validated our code with idealized test cases that have analytical solutions, including flow over a monochromatic wave with and without viscosity. The results showed very good agreement with analytical solutions for a monochromatic linear sine wave. Next, we presented test cases of turbulent flow over a moving sine wave at three different wave ages. The stress profiles showed expected decomposition between Reynolds stress, SGS stress, wave stress and pressure stress, and the total stress agrees with the theoretical profile for a pressuredriven channel flow. As further evidence for validation, we also compared the movingwave cases with physically equivalent cases from the original WRF–LES code. In the former, the wave propagates in the downwind direction, while in the latter the wave does not propagate, but the wind is adjusted so that the wind relative to the waves is the same as in the former. Results indicate that the mean streamwise velocity and Reynolds stress profiles for the two cases match. Additionally, we found that the results from WRF successfully capture the trend of wave growth rate as a function of wave age found in the literature.
In the future, this moving bottom implementation will make WRF a powerful tool to study wind–wave interactions. Moisture and temperature stratification are not considered in this paper, but as existing features in WRF they can be easily incorporated into moving bottom simulations. This paper focuses on idealized wave and wind conditions for the purpose of validation. As future work, the method can be extended to more realistic conditions such as simulating wind over seas or combined seas and swell, or assimilating measured meteorological data as forcing conditions. With the ability of WRF to incorporate realistic meteorological conditions, a more comprehensive understanding of the influence of waves on the ABL can be achieved. This implementation will also enable WRF to be coupled with wave models like HOS, and eventually be extended to simulate wind–wave interactions under realistic conditions, contributing to scientific and engineering applications from wave effect parameterizations to offshore wind resource assessment.
Acknowledgments.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DEAC5207NA27344, and supported by the Wind Energy Technologies Office. Support was also provided by the Office of Naval Research Grant N000142012707. We thank Dr. Peter P. Sullivan at the National Center for Atmospheric Research for his extensive help and insight.
Data availability statement.
All data and code used in this study will be available upon request.
REFERENCES
Aiyer, A. K., L. Deike, and M. E. Mueller, 2023: A sea surface–based drag model for largeeddy simulation of wind–wave interaction. J. Atmos. Sci., 80, 49–62, https://doi.org/10.1175/JASD210329.1.
BouZeid, E., C. Meneveau, and M. Parlange, 2005: A scaledependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys. Fluids, 17, 025105, https://doi.org/10.1063/1.1839152.
Buckley, M. P., and F. Veron, 2016: Structure of the airflow above surface waves. J. Phys. Oceanogr., 46, 1377–1397, https://doi.org/10.1175/JPOD150135.1.
Campbell, B. K., K. Hendrickson, and Y. Liu, 2016: Nonlinear coupling of interfacial instabilities with resonant wave interactions in horizontal twofluid plane Couette–Poiseuille flows: Numerical and physical observations. J. Fluid Mech., 809, 438–479, https://doi.org/10.1017/jfm.2016.636.
Cao, T., and L. Shen, 2021: A numerical and theoretical study of wind over fastpropagating water waves. J. Fluid Mech., 919, A38, https://doi.org/10.1017/jfm.2021.416.
Cao, T., B.Q. Deng, and L. Shen, 2020: A simulationbased mechanistic study of turbulent wind blowing over opposing water waves. J. Fluid Mech., 901, A27, https://doi.org/10.1017/jfm.2020.591.
Cao, T., X. Liu, X. Xu, and B. Deng, 2023: Investigation on mechanisms of fast opposing water waves influencing overlying wind using simulation and theoretical models. Phys. Fluids, 35, 015148, https://doi.org/10.1063/5.0132131.
Castorrini, A., S. Gentile, E. Geraldi, and A. Bonfiglioli, 2023: Investigations on offshore wind turbine inflow modelling using numerical weather prediction coupled with localscale computational fluid dynamics. Renewable Sustainable Energy Rev., 171, 113008, https://doi.org/10.1016/j.rser.2022.113008.
Cheung, T. K., and R. L. Street, 1988: The turbulent layer in the water at an air—Water interface. J. Fluid Mech., 194, 133–151, https://doi.org/10.1017/S0022112088002927.
Cimarelli, A., F. Romoli, and E. Stalio, 2023: On wind–wave interaction phenomena at low Reynolds numbers. J. Fluid Mech., 956, A13, https://doi.org/10.1017/jfm.2023.4.
D’Asaro, E., and C. McNeil, 2007: Air–sea gas exchange at extreme wind speeds measured by autonomous oceanographic floats. J. Mar. Syst., 66, 92–109, https://doi.org/10.1016/j.jmarsys.2006.06.007.
Deskos, G., S. Ananthan, and M. A. Sprague, 2022: Direct numerical simulations of turbulent flow over misaligned traveling waves. Int. J. Heat Fluid Flow, 97, 109029, https://doi.org/10.1016/j.ijheatfluidflow.2022.109029.
Di Bernardino, A., V. Mazzarella, M. Pecci, G. Casasanta, M. Cacciani, and R. Ferretti, 2022: Interaction of the sea breeze with the urban area of Rome: WRF mesoscale and WRF largeeddy simulations compared to groundbased observations. Bound.Layer Meteor., 185, 333–363, https://doi.org/10.1007/s10546022007345.
Dommermuth, D. G., and D. K. P. Yue, 1987: A highorder spectral method for the study of nonlinear gravity waves. J. Fluid Mech., 184, 267–288, https://doi.org/10.1017/S002211208700288X.
Druzhinin, O., Y. Troitskaya, W.T. Tsai, and P.C. Chen, 2019: The study of a turbulent air flow over capillarygravity water surface waves by direct numerical simulation. Ocean Model., 140, 101407, https://doi.org/10.1016/j.ocemod.2019.101407.
Grare, L., W. L. Peirson, H. Branger, J. W. Walker, J.P. Giovanangeli, and V. Makin, 2013: Growth and dissipation of windforced, deepwater waves. J. Fluid Mech., 722, 5–50, https://doi.org/10.1017/jfm.2013.88.
Hao, X., and L. Shen, 2019: Wind–wave coupling study using LES of wind and phaseresolved simulation of nonlinear waves. J. Fluid Mech., 874, 391–425, https://doi.org/10.1017/jfm.2019.444.
Hao, X., T. Cao, Z. Yang, T. Li, and L. Shen, 2018: Simulationbased study of windwave interaction. Procedia IUTAM, 26, 162–173, https://doi.org/10.1016/j.piutam.2018.03.016.
Hara, T., and P. P. Sullivan, 2015: Wave boundary layer turbulence over surface waves in a strongly forced condition. J. Phys. Oceanogr., 45, 868–883, https://doi.org/10.1175/JPOD140116.1.
Husain, N. T., T. Hara, and P. P. Sullivan, 2022a: Wind turbulence over misaligned surface waves and air–sea momentum flux. Part I: Waves following and opposing wind. J. Phys. Oceanogr., 52, 119–139, https://doi.org/10.1175/JPOD210043.1.
Husain, N. T., T. Hara, and P. P. Sullivan, 2022b: Wind turbulence over misaligned surface waves and air–sea momentum flux. Part II: Waves in oblique wind. J. Phys. Oceanogr., 52, 141–159, https://doi.org/10.1175/JPOD210044.1.
Hussain, A. K. M. F., and W. C. Reynolds, 1970: The mechanics of an organized wave in turbulent shear flow. J. Fluid Mech., 41, 241–258, https://doi.org/10.1017/S0022112070000605.
Kihara, N., H. Hanazaki, T. Mizuya, and H. Ueda, 2007: Relationship between airflow at the critical height and momentum transfer to the traveling waves. Phys. Fluids, 19, 015102, https://doi.org/10.1063/1.2409736.
Komori, S., R. Kurose, K. Iwano, T. Ukai, and N. Suzuki, 2010: Direct numerical simulation of winddriven turbulence and scalar transfer at sheared gas–liquid interfaces. J. Turbul., 11, N32, https://doi.org/10.1080/14685248.2010.499128.
Kumar, K., S. K. Singh, and L. Shemer, 2022: Directional characteristics of spatially evolving young waves under steady wind. Phys. Rev. Fluids, 7, 014801, https://doi.org/10.1103/PhysRevFluids.7.014801.
Kundu, P. K., I. M. Cohen, D. R. Dowling, and G. Tryggvason, 2016: Fluid Mechanics. 6th ed. Academic Press, 1002 pp.
Large, W. G., J. C. McWilliams, and S. C. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, 363–403, https://doi.org/10.1029/94RG01872.
Li, P. Y., D. Xu, and P. A. Taylor, 2000: Numerical modelling of turbulent airflow over water waves. Bound.Layer Meteor., 95, 397–425, https://doi.org/10.1023/A:1002677312259.
Li, T., and L. Shen, 2022a: Evolution of wave characteristics during windwave generation. Proc. 34th Symp. on Naval Hydrodynamics, Washington, DC, ONRS, 11 pp., https://arxiv.org/pdf/2208.02489.pdf.
Li, T., and L. Shen, 2022b: The principal stage in windwave generation. J. Fluid Mech., 934, A41, https://doi.org/10.1017/jfm.2021.1153.
Lin, M.Y., C.H. Moeng, W.T. Tsai, P. P. Sullivan, and S. E. Belcher, 2008: Direct numerical simulation of windwave generation processes. J. Fluid Mech., 616, 1–30, https://doi.org/10.1017/S0022112008004060.
Liu, S., A. Kermani, L. Shen, and D. K. P. Yue, 2009: Investigation of coupled airwater turbulent boundary layers using direct numerical simulations. Phys. Fluids, 21, 062108, https://doi.org/10.1063/1.3156013.
Mirocha, J. D., J. K. Lundquist, and B. Kosović, 2010: Implementation of a nonlinear subfilter turbulence stress model for largeeddy simulation in the advanced research WRF Model. Mon. Wea. Rev., 138, 4212–4228, https://doi.org/10.1175/2010MWR3286.1.
Moeng, C.H., J. Dudhia, J. Klemp, and P. Sullivan, 2007: Examining twoway grid nesting for large eddy simulation of the PBL using the WRF Model. Mon. Wea. Rev., 135, 2295–2311, https://doi.org/10.1175/MWR3406.1.
Patton, E. G., P. P. Sullivan, B. Kosović, J. Dudhia, L. Mahrt, M. Žagar, and T. Marić, 2019: On the influence of swell propagation angle on surface drag. J. Appl. Meteor. Climatol., 58, 1039–1059, https://doi.org/10.1175/JAMCD180211.1.
Plant, W. J., 1982: A relationship between wind stress and wave slope. J. Geophys. Res., 87, 1961–1967, https://doi.org/10.1029/JC087iC03p01961.
Reichl, B. G., T. Hara, and I. Ginis, 2014: Sea state dependence of the wind stress over the ocean under hurricane winds. J. Geophys. Res. Oceans, 119, 30–51, https://doi.org/10.1002/2013JC009289.
Skamarock, W. C., and Coauthors, 2008: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN475+STR, 113 pp., https://doi.org/10.5065/D68S4MVH.
Skamarock, W. C., and Coauthors, 2019: A description of the Advanced Research WRF Model version 4. NCAR Tech. Note NCAR/TN556+STR, 145 pp., https://doi.org/10.5065/1dfh6p97.
Sullivan, P. P., J. C. McWilliams, and C.H. Moeng, 2000: Simulation of turbulent flow over idealized water waves. J. Fluid Mech., 404, 47–85, https://doi.org/10.1017/S0022112099006965.
Sullivan, P. P., J. B. Edson, T. Hristov, and J. C. McWilliams, 2008: Largeeddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves. J. Atmos. Sci., 65, 1225–1245, https://doi.org/10.1175/2007JAS2427.1.
Sullivan, P. P., J. C. McWilliams, and E. G. Patton, 2014: Largeeddy simulation of marine atmospheric boundary layers above a spectrum of moving waves. J. Atmos. Sci., 71, 4001–4027, https://doi.org/10.1175/JASD140095.1.
Sullivan, P. P., M. L. Banner, R. P. Morison, and W. L. Peirson, 2018a: Impacts of wave age on turbulent flow and drag of steep waves. Procedia IUTAM, 26, 174–183, https://doi.org/10.1016/j.piutam.2018.03.017.
Sullivan, P. P., M. L. Banner, R. P. Morison, and W. L. Peirson, 2018b: Turbulent flow over steep steady and unsteady waves under strong wind forcing. J. Phys. Oceanogr., 48, 3–27, https://doi.org/10.1175/JPOD170118.1.
Talbot, C., E. BouZeid, and J. Smith, 2012: Nested mesoscale largeeddy simulations with WRF: Performance in real test cases. J. Hydrometeor., 13, 1421–1441, https://doi.org/10.1175/JHMD11048.1.
Wicker, L. J., and W. C. Skamarock, 2002: Timesplitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, https://doi.org/10.1175/15200493(2002)130<2088:TSMFEM>2.0.CO;2.
Wu, J., and L. Deike, 2021: Wind wave growth in the viscous regime. Phys. Rev. Fluids, 6, 094801, https://doi.org/10.1103/PhysRevFluids.6.094801.
Wu, J., S. Popinet, and L. Deike, 2022: Revisiting wind wave growth with fully coupled direct numerical simulations. J. Fluid Mech., 951, A18, https://doi.org/10.1017/jfm.2022.822.
Xiao, S., and D. Yang, 2019: Largeeddy simulationbased study of effect of swellinduced pitch motion on wakeflow statistics and power extraction of offshore wind turbines. Energies, 12, 1246, https://doi.org/10.3390/en12071246.
Yang, D., and L. Shen, 2009: Characteristics of coherent vortical structures in turbulent flows over progressive surface waves. Phys. Fluids, 21, 125106, https://doi.org/10.1063/1.3275851.
Yang, D., and L. Shen, 2011a: Simulation of viscous flows with undulatory boundaries. Part I: Basic solver. J. Comput. Phys., 230, 5488–5509, https://doi.org/10.1016/j.jcp.2011.02.036.
Yang, D., and L. Shen, 2011b: Simulation of viscous flows with undulatory boundaries. Part II: Coupling with other solvers for twofluid computations. J. Comput. Phys., 230, 5510–5531, https://doi.org/10.1016/j.jcp.2011.02.035.
Yang, D., C. Meneveau, and L. Shen, 2013: Dynamic modelling of seasurface roughness for largeeddy simulation of wind over ocean wavefield. J. Fluid Mech., 726, 62–99, https://doi.org/10.1017/jfm.2013.215.
Yang, D., C. Meneveau, and L. Shen, 2014a: Effect of downwind swells on offshore wind energy harvesting—A largeeddy simulation study. Renewable Energy, 70, 11–23, https://doi.org/10.1016/j.renene.2014.03.069.
Yang, D., C. Meneveau, and L. Shen, 2014b: Largeeddy simulation of offshore wind farm. Phys. Fluids, 26, 025101, https://doi.org/10.1063/1.4863096.
Zhang, W.Y., W.X. Huang, and C.X. Xu, 2019: Very largescale motions in turbulent flows over streamwise traveling wavy boundaries. Phys. Rev. Fluids, 4, 054601, https://doi.org/10.1103/PhysRevFluids.4.054601.