## 1. Introduction

*M*profile, which is defined by the following empirical equation (Bean and Dutton 1966):

*T*is the air temperature (K),

*e*and

*P*denote the water vapor and the total atmospheric pressure (mb; 1 mb = 1 hPa), and

*z*is the height above sea surface. According to (1),

*M*tends to linearly increase with the vertical distance

*z*in the troposphere with exceptions such as across a strong inversion or in a marine surface layer. In a marine surface layer (i.e., a few to dozens of meters deep),

*M*may decrease with the height associated with the decay of the vapor pressure

*e*before it increases farther aloft, and radio waves tend to downward refract. The ED top is located at the turning level, where

*dM*/

*dz*= 0, and the ED characteristics are determined by meteorological conditions and interactions between the marine surface layer and ocean.

The *M* profile in a surface layer is strongly dependent on the temperature and specific humidity profiles that are usually obtained using the Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954) in electromagnetic (EM) propagation models (Babin et al. 1997). According to MOST, nondimensional gradients of mean winds and scalars in a surface layer are universal functions of a nondimensional height (i.e., referred to as flux relationships) over a homogeneous and stationary surface. Over the past 60 years, MOST has been tested extensively using observations over homogenous land surfaces (e.g., Businger et al. 1971), based on which multiple universal functions had been proposed. The adequacy of these empirical universal functions for a marine surface layer has not been thoroughly evaluated. Limited existing observations over sea are somewhat controversial. Measurements over the open ocean from the Equatorial Pacific Gas Exchange Study 2001 (GasEx-2001; Edson et al. 2004; McGillis et al. 2004) found reasonable agreement between the measured scalar profiles above the wave boundary layer and the corresponding MOST predictions. On the other hand, the homogeneous and stationary surface assumptions for MOST apparently become problematic over propagating waves, coastal oceans, or ocean fronts (e.g., Mahrt and Khelif 2010; Mahrt et al. 2014). For example, both observations (e.g., Rutgersson et al. 2001; Smedman et al. 2009) and large-eddy simulations (LES) (e.g., Sullivan et al. 2008) documented the formation of a low-level wind jet associated with near-zero or positive momentum flux in the ambient wind direction under swell conditions. Observations obtained from the Rough Evaporation Duct (RED) experiment suggested that waves play a role in regulating scalar profiles in the surface layer, and the normalized scalar gradients depart from those universal stability functions derived from observations over land (Anderson et al. 2004). Based on analysis of tower observations, Smedman et al. (2007) found that the similarity relationship for temperature breaks down in a weakly unstable regime when the Obukhov length is less than −150 m. Impact of breaking waves on vertical scalar transport was briefly discussed in Sullivan et al. (2018) based on a pair of large-eddy simulations. They demonstrated that above the wave boundary layer, to a good approximation, the domain-averaged scalar and wind speed profiles exhibit logarithmic distributions.

The atmospheric boundary layer over swell waves has been subject to several LES studies over the past decade (Sullivan et al. 2008, 2014, 2018; Nilsson et al. 2012; Jiang et al. 2016), with emphasis on momentum fluxes, form drag and low-level wind jet dynamics under neutral to convective conditions. The impact of swell waves on scalar profiles has received little attention. The main objective of this study is to explore the influence of swell on scalar profiles in a marine surface layer and the underlying dynamics under different stability conditions using surface-layer-resolving large-eddy simulations. The remainder of this paper is organized as follows. The LES code and configuration are illustrated in section 2. The LES results are presented in sections 3 and 4, with emphasis on the stable and unstable marine boundary layer, respectively. The eddy diffusivities from LES simulations and MOST predictions are compared in section 5. Section 6 contains discussion and concluding remarks.

## 2. Large-eddy simulation code and configuration

There are seven prognostic variables in the above equations, namely, the three Cartesian velocity components, *u*_{i} = (*u*, *υ*, *w*), potential temperature *θ*, the air density–normalized perturbation pressure, *p** = *p*/*ρ*_{0} (*p* is the perturbation pressure and *ρ*_{0} is the air density), specific humidity *q*, and the subgrid-scale (SGS) turbulence kinetic energy *e*. The other symbols include the density-normalized external (i.e., synoptic scale or mesoscale) pressure gradient, (1/*ρ*_{0})(∂*P*/∂*x*_{i}) = (−*fV*_{g}, *fU*_{g,} 0), which is in balance with the Coriolis force, the Coriolis coefficients *f*_{i} = (0, 0, *f*), reference potential temperature *θ*_{0}, virtual potential temperature, *θ*_{υ} = *θ*(1 + 0.61*q*), strain rates, *S*_{ij} = (∂*u*_{i}/∂*x*_{j} + ∂*u*_{j}/∂*x*_{i})/2, dissipation rate *ε*, and the SGS momentum, potential temperature, and water vapor fluxes, *τ*_{ij}, *τ*_{iθ}, and *τ*_{iq}. The SGS fluxes and dissipation rate are parameterized following Deardorff (1972) and Moeng and Wyngaard (1988). A monochromatic wave is included along the surface as a proxy for swell, which can be written as *h*(*x*, *t*) = (*s*/*k*) sin(*kx* − *ct*), where *h*(*x*, *t*) denotes the vertical displacement of a wavy surface, *k* = 2*π*/*λ* is the wavenumber, *λ* is the wavelength, *s* = *ka* is the wave slope, and *a* is the wave maximum height. The swell propagates at the phase speed of a linear deep wave,

*x*

_{i}= (

*x*,

*y*,

*z*) are mapped to the computational coordinates

*ξ*

_{i}= (

*ξ*,

*η*,

*ζ*) through a curvilinear transformation (e.g., Sullivan et al. 2014):

*H*is the model-top level. The transformed LES equations in the computational coordinates can be found in Sullivan et al. (2014).

The horizontal derivatives in (2a)–(2e) are evaluated using a pseudospectral approximation and the vertical derivatives are obtained using a second-order finite-difference scheme. These equations are integrated in time using the fully explicit third-order Runge–Kutta scheme (e.g., Sullivan et al. 2008) with a dynamic time interval adjusted for a given maximum Courant–Friedrichs–Lewy (CFL) number of 0.3 at each time step.

### Model configuration and simulations

There are 160 vertical levels with a flat model top located at *H* = 768 m (unless specified otherwise), where a radiation boundary condition is applied to minimize the downward reflection of internal gravity waves (Klemp and Durran 1983). The vertical grid spacing slowly increases with the height with the ratio between any two adjacent cell held as a constant, that is, Δ*z*_{i+1}/Δ*z*_{i} ~ 1.01. The horizontal domain has 256 × 256 grid points with periodic conditions applied along the sidewalls. For the simulations in groups A–E, the horizontal grid spacing is Δ*x* = Δ*y* = 3 m (Table 1). According to Jiang et al. (2018), for a given horizontal grid space Δ*x*, dominant eddies are underresolved in a layer approximately below 2.5Δ*x* ~ 7.5 m where the contribution to the SGS fluxes accounts for 10% or more (referred to as SGS buffer layer). In a SGS buffer layer, the simulated vertical gradients of mean winds and scalars are sensitive to the SGS model and the “overshoot” problem (i.e., a suspicious maximum in the simulated gradients) described by Mason and Thomson (1992) may exist.

The control parameters and model grids for the six groups of simulations. Columns 2–7 correspond to the geostrophic wind speed *U*_{g}, initial air–sea temperature difference Δ*T*_{as}, the saturation specific humidity *q*_{s}, horizontal grid spacing Δ_{h}, the depth of the model domain *H*, and the surface virtual potential temperature flux

The LES model is initialized with a uniform longitudinal wind speed, *U*(*z*) = *U*_{g}, with the large-scale pressure gradient in balance with the Coriolis force (i.e., *dP*/*dy* = −*fU*_{g}, where the Coriolis coefficient *f* = 10^{−4} s^{−1}). The potential temperature is constant, *θ*(*z*) = *θ*_{0}, below 200 m and linearly increase with the altitude aloft as *θ*(*z*) = *θ*_{0} + 0.003(*z* − 200), corresponding to a buoyancy frequency of *q*(*z*) = 5 g kg^{−1}. Results from six groups of simulations are presented, corresponding to stable (groups A–C) and unstable (groups D–F) boundary layers, respectively. Each group includes four simulations with identical model configuration except for the swell. These simulations are referred to as FLAT (i.e., no swell), WFLG-64, WFLG-128 (wind-following swell with wavelength *λ* = 64 and 128 m), and WOPS or WOPS-64 (wind-opposing swell with *λ* = 64 m).

**u**

_{1},

**v**

_{1}) is the wind vector at the first model level,

*s*

_{1}and

*s*

_{s}denote scalar

*s*values (i.e., temperature or specific humidity) at the first model level and the surface, respectively. Here

*C*

_{D},

*C*

_{θ}, and

*C*

_{q}are the bulk aerodynamic coefficients (often referred to as the drag coefficient, Stanton number and Dalton number, respectively), evaluated at each horizontal grid point following the Monin–Obukhov similarity theory using a constant roughness length (i.e.,

*z*

_{0}=

*z*

_{θ0}=

*z*

_{q0}= 0.0002 m, where

*z*

_{0},

*z*

_{θ0}, and

*z*

_{q0}denote the roughness length for momentum, temperature and moisture, respectively). More details about the stress and flux calculation over the wavy surface can be found in appendix b of Sullivan et al. (2014).

The mean profiles shown in this study are spatially averaged over the horizontal domain and temporally between 5 and 5.5 h. We conclude that our simulations reach a statistical equilibrium state for *T* > 5 h based on inspection of time series of the domain-averaged boundary layer (BL) parameters and time-scale analysis. For each swell simulation, the swell is introduced at *T* = 2 h and rapid adjustment takes place between 2 and 3 h. At *T* = 5 h, the domain-averaged parameters such as surface fluxes, turbulence kinetic energy and boundary layer depth evolve rather slowly with time (see examples shown in Fig. 1). The turbulence adjustment time (or large-eddy turnover time) can be written as *T*_{t} = *Z*_{i}/*u*_{*} for a stable BL or *Z*_{i} is the BL depth and

## 3. Influence of swell on a stable boundary layer

To examine the impact of swell on a stable boundary layer (SBL) under different meteorological conditions, three groups of simulations have been conducted (i.e., groups A–C; see Table 1 for control parameters). The BL is considered to be stably stratified when the surface virtual temperature flux, *τ*_{sh} is the air density normalized shear stress, *κ* = 0.4 is the von Kármán constant, *g* is gravity acceleration, and

The control and derived parameters from groups A, B, and C. Columns 3–8 correspond to the shear stress–based friction velocity (m s^{−1}), normalized pressure drag ^{−2}), latent heat flux (^{−2}), Obukhov length *L* (m), and the evaporation duct height *H*_{ED} (m), respectively.

### a. Influence of swell on stable boundary layer characteristics

We start by visually inspecting the mean profiles (Fig. 2) and vertical cross-section snapshots (Fig. 3) from the group A simulations. The geostrophic wind speed *U*_{g} for group A is 5 m s^{−1}, and the initial air–sea temperature difference Δ*T*_{as} = *T*_{air} − SST = 5 K, where the sea surface temperature, SST = 298.16 K, is held constant. For these simulations, the sensible heat flux is negative, the latent heat flux is positive, and the resulting Obukhov length is positive (Table 2).

Vertical profiles of (a) the *u*-component of the horizontal wind *U* (m s^{−1}), (b) potential temperature *θ* − *θ*_{0} (K), (c) specific humidity *q* (g kg^{−1}), and (d) modified refractivity *M* derived from the group A simulations. Only the lowest 300 m are shown.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of (a) the *u*-component of the horizontal wind *U* (m s^{−1}), (b) potential temperature *θ* − *θ*_{0} (K), (c) specific humidity *q* (g kg^{−1}), and (d) modified refractivity *M* derived from the group A simulations. Only the lowest 300 m are shown.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of (a) the *u*-component of the horizontal wind *U* (m s^{−1}), (b) potential temperature *θ* − *θ*_{0} (K), (c) specific humidity *q* (g kg^{−1}), and (d) modified refractivity *M* derived from the group A simulations. Only the lowest 300 m are shown.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical cross sections of (left) potential temperature *θ* − *θ*_{0} (K), (middle) *u* wind component (m s^{−1}), and (right) vertical gradient of specific humidity *dq*/*dz* (g kg^{−1} km^{−1}) derived from simulations (a)–(c) A_{0} (FLAT), (d)–(f) A_{1} (WFLG-64), and (g)–(i) A_{3} (WOPS-64).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical cross sections of (left) potential temperature *θ* − *θ*_{0} (K), (middle) *u* wind component (m s^{−1}), and (right) vertical gradient of specific humidity *dq*/*dz* (g kg^{−1} km^{−1}) derived from simulations (a)–(c) A_{0} (FLAT), (d)–(f) A_{1} (WFLG-64), and (g)–(i) A_{3} (WOPS-64).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical cross sections of (left) potential temperature *θ* − *θ*_{0} (K), (middle) *u* wind component (m s^{−1}), and (right) vertical gradient of specific humidity *dq*/*dz* (g kg^{−1} km^{−1}) derived from simulations (a)–(c) A_{0} (FLAT), (d)–(f) A_{1} (WFLG-64), and (g)–(i) A_{3} (WOPS-64).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Over a flat surface, a shallow SBL develops, associated with surface cooling (Fig. 2a), with turbulent eddies largely confined in the lowest ~60 m (Figs. 3a–c). The SBL is relatively uniformly stratified with pronounced vertical wind shear concentrated in the lowest ~20 m (Figs. 2a,b and 3a–c). Finescale forward-tilting filament-like patterns are evident approximately between 10 and 60 m in the specific humidity gradient (Fig. 3c) and vertical temperature gradient (not shown) fields. These “filaments,” often referred to as temperature fronts, microscale fronts, or cliff–ramp fronts, in stable boundary layers have been documented by field observations (Thorpe and Hall 1980), wind tunnel experiments (Chen and Blackwelder 1978), direct numerical simulations (Chung and Matheou 2012), and high-resolution large-eddy simulations (Sullivan et al. 2016). Over WFLG swell (i.e., A1), the SBL becomes substantially shallower than over a flat surface. The potential temperature (specific humidity) profile changes from nearly linearly increasing (decreasing) in FLAT to being concave down with slopes much reduced in the lowest 15–20 m and increased up to the top of the BL (Figs. 2 and 3d–f). Vertical oscillations with a wavelength comparable to the swell wavelength are evident in both the scalar and wind vertical cross sections, suggesting that the direct impact of swell extending throughout the depth of the SBL. This implies that for the two WFLG simulations the wave boundary layer (WBL) depth is comparable to the SBL depth, in contrast to the conventional notion that WBL only accounts for the lowest *O*(1) m. For scalars, the vertical oscillations are more pronounced near the SBL top where their vertical gradients are large and swell-induced vertical velocity is still substantial. For the horizontal winds the sinusoidal variation is characterized by maxima at the surface, which decays with the vertical distance. Microfronts are also visible in scalar gradient fields in the WFLG simulations similar to those in the FLAT simulation.

Compared to FLAT, the wind speed in the BL is noticeably enhanced in WFLG-64 (Fig. 2a), and the surface shear stress increases accordingly (Table 2). The wind speed in the SBL becomes even stronger in WFLG-128 with a low-level wind maximum located at approximately 30 m above the sea surface [referred to as a wave-driven wind jet (WDWJ hereafter); see section 3b for discussion of WDWJ dynamics]. The swell also has significant impact on the mean scalar profiles (Figs. 2b,c). Consequently, the modified refractivity (*M*) profile exhibits high sensitivity to the underlying swell. For FLAT, the *M* profile shows a maximum at the sea level, a minimum near the BL top and an approximately linear decrease between the two, implying the existence of a surface duct for radio waves with its height comparable to the SBL height. Over WFLG swell, in accordance with variations in the temperature and specific humidity profiles, the *M* profiles in the SBL change substantially, characterized by a larger *M* layer below ~20 m and a rapid decrease in the upper SBL. Compared with FLAT, the surface duct in the two simulations with WFLG swell is lower and stronger (note that ED strength is measured by the difference between *M* at the surface and at the ED top where *M* reaches a minimum). It is worth noting that the Obukhov length from the two simulations with WFLG swell increases by a factor of 5 or more, implying that the surface layer in WFLG-64 and WFLG-128 is less stable than in FLAT. This is consistent with the much reduced potential temperature and specific humidity gradients in the shallow layer immediately above the wavy surface in these two WFLG simulations. Conventionally, the term surface layer is used for the lowest 10% of a boundary layer or a constant flux layer above the surface. Here we use the term rather loosely to refer to the lowest ~30 m of a stable BL (i.e., approximately 10% of the deepest BL depth in groups A and B) and ~50 m of a convective BL layer (i.e., approximately 10% of the deepest BL depth in groups D and E) for the convenience of discussion.

The WOPS swell does exactly the opposite of what WFLG swell does, namely, reducing the wind speed in the BL, increasing the BL depth, and spreading the vertical wind shear over a much deeper layer (Figs. 2 and 3g–i). The SBL depth for WOPS-64 is *Z*_{i} ~ 275 m, which is 4 times or more of the SBL depth in FLAT and WFLG simulations. Sinusoidal variations are noticeable only in the lowest ~20 m. Filament patterns are evident in the moisture gradient field, but with gentler slopes, consistent with the reduced stratification [i.e., smaller *M* is smaller near the surface, associated with smaller specific humidity, and slowly increases with height aloft, with a minimum (i.e., ED top) at ~46 m.

*θ*(

*z*) −

*θ*

_{0}= (

*θ*

_{s}−

*θ*

_{0})(1 −

*z*/

*Z*

_{i}), and other parameters such as

*C*

_{θ}, SST, and

*U*are constant in time, the air–sea temperature difference at time

*T*can be written as

*T*

_{as}denotes the initial air–sea temperature difference. Similarly, the evolution of specific humidity near the surface can be written as

*q*

_{s}(

*T*) denotes the air specific humidity near the surface and

*q*

_{0}is the saturation specific humidity for a given SST. According to (6), the air–sea temperature difference decreases exponentially with time with an

*e*-folding time scale of

*T*

_{0}=

*Z*

_{i}/(2

*C*

_{q}

*U*). Over WFLG swell, the BL is shallower and surface wind speed is stronger, and the decrease of the air–sea temperature difference is faster than over a flat surface, although this trend is partially counterbalanced by the concave down potential temperature profile near the surface. Similarly, the latent heat flux at

*T*= 5 h is smaller over WFLG swell due to a smaller air–sea specific humidity difference. For the same reason, over WOPS swell, both sensible and latent heat fluxes are larger and the resulting Obukhov length is substantially smaller than their counterparts over a flat surface.

### b. Fluxes, turbulence, and dynamics

According to previous studies (e.g., Hanley and Belcher 2008; Sullivan et al. 2008; Semedo et al. 2009), under low to moderate wind conditions, WFLG swell tends to accelerate the airflow and promote the formation of a WDWJ by “pumping” momentum into the atmospheric flow immediately above the wavy surface through forward-pointing pressure drag (i.e., shear stress and form drag have opposite signs). The pressure drag and total stress (i.e., shear stress + pressure drag) profiles are shown in Fig. 4. For WFLG-64, pressure drag reduces the total stress substantially throughout the BL and for WFLG-128, the total stress changes from negative to positive as the pressure drag becomes dominant. Over WOPS swell, both shear stress and pressure drag are negative, resulting substantially larger total negative stress. As demonstrated in Jiang et al. (2016), the formation of WDWJ in a neutral MBL over swell occurs as the total surface stress [i.e., *τ* = *τ*_{sh} + *D*_{0}, where the pressure drag over monochromatic swell with wavelength *λ* is given by

Vertical profiles of (a) total momentum flux along the geostrophic wind direction *F*_{x} (sum of shear stress and pressure drag; m^{2} s^{−2}), (b) pressure drag (m^{2} s^{−2}), and (c) the moist buoyancy frequency squared [^{−2}] derived from the group A simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of (a) total momentum flux along the geostrophic wind direction *F*_{x} (sum of shear stress and pressure drag; m^{2} s^{−2}), (b) pressure drag (m^{2} s^{−2}), and (c) the moist buoyancy frequency squared [^{−2}] derived from the group A simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of (a) total momentum flux along the geostrophic wind direction *F*_{x} (sum of shear stress and pressure drag; m^{2} s^{−2}), (b) pressure drag (m^{2} s^{−2}), and (c) the moist buoyancy frequency squared [^{−2}] derived from the group A simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Shown in Fig. 4c are the moist buoyancy-frequency-squared profiles, showing that WFLG swell reduces the stability in the lowest ~15 m and increase the stability aloft. WOPS swell does the opposite; increasing the stability in the lowest ~10 m and decreasing the stability aloft. The turbulence kinetic energy (TKE) profile is sensitive to swell characteristics (Fig. 5a). It is worth noting that the large velocity variances immediately above the surface for swell simulations include significant contribution from swell induced perturbations, which decay rapidly with the vertical distance from the surface (e.g., Jiang et al. 2016). Above the surface layer, TKE from WFLG swell simulations is sizably weaker than in FLAT, which is consistent with the variation of the vertical shear. This can be seen clearly in the TKE shear production rate [i.e., SPR; defined as

Vertical profiles of (a) resolved turbulence kinetic energy (TKE; m^{2} s^{−2}), (b) shear production rate of turbulence (SPR; m^{2} s^{−3}), and (c) buoyancy production rate (BPR; m^{2} s^{−3}) derived from the group A simulations. TKE and SPR are shown in logarithmic coordinates.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of (a) resolved turbulence kinetic energy (TKE; m^{2} s^{−2}), (b) shear production rate of turbulence (SPR; m^{2} s^{−3}), and (c) buoyancy production rate (BPR; m^{2} s^{−3}) derived from the group A simulations. TKE and SPR are shown in logarithmic coordinates.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of (a) resolved turbulence kinetic energy (TKE; m^{2} s^{−2}), (b) shear production rate of turbulence (SPR; m^{2} s^{−3}), and (c) buoyancy production rate (BPR; m^{2} s^{−3}) derived from the group A simulations. TKE and SPR are shown in logarithmic coordinates.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

### c. Groups B and C

To explore the influence of swell on a SBL over a broader parameter space, two additional groups of simulations (i.e., B and C) have been conducted. The model configuration for group B is identical to group A except that the surface saturation specific humidity (*q*_{s} = 15 g kg^{−1}) is lower, corresponding to a lower SST (i.e., 293.6 K) and the initial air–sea temperature difference is smaller (i.e., Δ*T*_{as} = 2 K). Associated with the changes in the sensible and latent heat fluxes, the Obukhov length for B0 is one order of magnitude larger than that for A0, indicative of a much less stable BL (i.e., closer to neutral) in B0. Consequently, the BL in B0 is nearly 3 times as deep as in A0 (Fig. 6, Table 2). Despite the substantial differences in the BL depth and stability between A0 and B0, their responses to swell forcing are rather similar. Specifically, the wind speed in the boundary layer is boosted by WFLG swell in WFLG-64, and a wind jet forms in WFLG-128 in accordance to the normalized pressure drag becoming less than −1 (Table 2). The potential temperature profile from B0 exhibits a two-layer structure with a more stable layer below ~20 m and a less stable layer approximately between 20 and 175 m (Fig. 6b). Over WFLG swell, the potential temperature profiles are characterized by a three-layer structure in the BL, namely a shallow less stable layer (~15 m) immediately above the surface, a more stable middle layer, and then another less stable layer aloft (i.e., ~above 60 m). The less stable layer above the surface over WFLG swell is consistent with the substantial increase in the Obukhov length (Table 2). The specific humidity profiles exhibit similar layered structure as the potential temperature except that the specific humidity decreases with height. Over WOPS swell, an approximately 15-m-thick more stable layer is present over the surface, above which a deep nearly well-mixed layer extends to ~275 m (Fig. 6b), in accordance with much stronger turbulence (not shown) than in the FLAT and WFLG swell simulations. An evaporation duct forms with *h*_{ED} ~ 14.3 m, which is substantially lower than its counterpart in other B group simulations.

As in Fig. 2, but for the group B simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for the group B simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for the group B simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

The model configuration for group C is identical to group B except that the geostrophic wind speed is doubled, and accordingly the friction velocity and BL depth are substantially larger than their counterpart in group B (Fig. 7 and Table 2). The Obukhov length for C0 is larger than in B0, suggesting that the boundary layer in C0 is even closer to neutral than that in B0. Similar response of the wind profiles to swell forcing to groups B and C is evident (Fig. 7), but much weaker. For example, the difference in the mean wind profiles between FLAT, WFLG-64, and WFLG-128 is rather insignificant (Fig. 7a). This is consistent with the small stress-normalized pressure drag (Table 2) in group C simulations. The pressure drag plays a much less significant role in modifying the vertical wind shear and accordingly the turbulence mixing in the MBL. Still, the rather minor changes in the temperature and moisture profiles near the surface lead to a nonnegligible increase in the evaporation duct height from FLAT (Table 2). The impact of WOPS swell on the mean profiles is more pronounced, being consistent with its much larger stress-normalized pressure drag. Consequently, the wind speed is reduced and the vertical shear of the horizontal winds is spread out to the middle and upper portion of the MBL. The vertical redistribution of the wind shear presumably has impact on the turbulence mixing of scalars (i.e., potential temperature and specific humidity), and accordingly, the scalar and *M* profiles exhibit noticeable changes from FLAT. The resulted ED height is more than 10 m lower than FLAT.

As in Fig. 2, but for the group C simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for the group C simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for the group C simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

In summary, swell may exert significant influence on the wind and scalar profiles in a stable marine boundary layer, leading to dramatic change in evaporation duct characteristics. In general, the swell impact is more significant on a more stable boundary layer with lower winds. Quantitatively, the impact from swell varies substantially with swell characteristics such as wavelength, swell–wind alignment, and presumably wave slope.

## 4. Impact of swell on unstable BL

To explore the response of an unstable BL to swell forcing, results from three groups of simulations (i.e., D, E, F, and see Table 1 for control parameters) of unstable BLs are analyzed in this section with more emphasis on group D.

### a. Influence of swell on unstable boundary layer characteristics

The model configuration for group D is identical to the stable group B except that the SST in group D is warmer. As the SST is still initially 2 K cooler than the airflow above the sea surface, the sensible heat flux remains negative in group D. However, the surface virtual temperature flux becomes positive (i.e., the Obukhov length becomes negative) owing to a larger latent heat flux (Table 3), resulting in a number of substantial changes in the BL characteristics from their stable counterparts (i.e., group B). Specifically, (i) the BL becomes much deeper with an inversion located between 300 and 400 m (Fig. 8), (ii) a well-mixed layer forms between a shallow layer above the surface (below ~20 m) and the BL top inversion (~300 m), and (iii) an ED forms much closer to the sea surface and an elevated duct forms near the BL top inversion.

As in Fig. 2, but for group D simulations. The profiles for the lowest 500 m are shown.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for group D simulations. The profiles for the lowest 500 m are shown.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for group D simulations. The profiles for the lowest 500 m are shown.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

The impact of swell on the BL characteristics in group D is qualitatively similar to their counterparts in group B in the following aspects. First, the BL tends to become shallower over WFLG swell and deeper over WOPS swell. Second, the wind speed in the BL becomes stronger over WFLG swell with enhanced shear in a shallow (~20 m deep) layer immediately above the surface (Fig. 9a). With the stress-normalized pressure drag less than −1, a wind maximum (located at ~30 m; Fig. 8a) appears in the WFLG-128 simulation. Third, the BL winds become substantially weaker over WOPS swell with the surface shear layer expanding significantly (~100 m; Fig. 8a). Finally, the BL appears to be more sensitive to WOPS than WFLG swell. On the other hand, compared to their counterparts in group B, the impact of swell on the unstable BL in group D is much less pronounced. For example, the decrease of BL depth due to WFLG swell is less than 5% and the increase of BL depth over WOPS swell is approximately 15%. The sinusoidal patterns penetrating the whole stable BL over WFLG swell (Fig. 3d) are barely seen in the corresponding potential temperature cross section in group D (Fig. 9d). The sinusoidal undulation is distinguishable in the vertical cross section of horizontal winds, but only in the lowest ~15 m (Fig. 9e). The wavy vertical displacement of the inversion has a characteristic wavelength of ~300–500 m, likely associated with atmospheric internal waves as opposed to the swell wave in the stable BL. For group D, in addition to an evaporation duct in the surface layer, an elevated duct appears below the MBL top associated with a sharp decrease in the specific humidity. Both the ED and elevated duct heights vary substantially with the underlying swell. For example, the ED height increases by ~8 m over the longer WFLG swell (i.e., WFLG-128 m), and decreases by ~10 m over the WOPS swell.

As in Fig. 3, but for group D simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 3, but for group D simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 3, but for group D simulations.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Turbulence tends to become weaker above the surface layer over WFLG swell (Fig. 10). The weakening of turbulent eddies between 20 and 100 m over WFLG swell is also evident in the specific humidity gradient cross sections (i.e., cf. Figs. 9c and 9f). The shear production rate of TKE shows a moderate decrease above the lowest 20 m over WFLG swell (Fig. 10b). For simulations over a flat surface or WFLG swell, the buoyancy production rate (BPR) is positive over the lower two-thirds of the BL and becomes negative in the top one-third associated with downward entrainment of warmer air near the inversion level. Between approximately 30–200 m, the magnitude of the BPR is comparable to or larger than the corresponding shear production rate, suggesting the importance of buoyant production of turbulence even in this weakly unstable group.

As in Fig. 5, but derived from group D. The horizontal axes are in logarithmic coordinates.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 5, but derived from group D. The horizontal axes are in logarithmic coordinates.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 5, but derived from group D. The horizontal axes are in logarithmic coordinates.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Comparing to over WLFG swell, the weakly unstable BL in group D over WOPS swell exhibits more substantial changes. For example, in the middle of the BL (~150 m), the TKE over WOPS swell is approximately 4 times of its counterpart over a flat surface, and the shear production rate is about 3–5 times larger than its counterpart in the other three simulations, while the BPR is sizably smaller.

### b. Groups E and F

Two additional groups of unstable BL simulations have been conducted with lower geostrophic wind speed (i.e., group E) or negative air–sea temperature difference (i.e., group F), respectively (Table 1). The results are summarized in Figs. 11 and 12 and Table 3.

As in Fig. 2, but for the group E simulations. The wind maximum is located at 8.5 and 11 m above the sea surface for WFLG-64 and WFLG-128, respectively.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for the group E simulations. The wind maximum is located at 8.5 and 11 m above the sea surface for WFLG-64 and WFLG-128, respectively.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for the group E simulations. The wind maximum is located at 8.5 and 11 m above the sea surface for WFLG-64 and WFLG-128, respectively.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for group F.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for group F.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 2, but for group F.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Compared to group D, the geostrophic wind speed in group E is lower (i.e., *U*_{g} = 2.24 m s^{−1} instead of 5 m s^{−1}). Accordingly, the friction velocity and the sensible heat flux are smaller (Table 3). In terms of response to swell forcing, the influence of swell on group E is obviously more significant. Due to the lower wind speed, the stress-normalized pressure drag is greater than unity in the two WFLG simulations, and consequently, a wind maximum (i.e., WDWJ) appears near the surface. The wind maximum level and jet strength increase with the swell wavelength. The enhancement of the low-level wind speed over WFLG swell increases the magnitude of the sensible and latent heat fluxes, which lead to a colder and moister surface layer (Figs. 11b,c). The evaporation duct height increases accordingly (by ~1.5–5 m, Fig. 11d and Table 3). The impact of WOPS swell on the wind and scalar profiles is qualitatively similar to that in group D and the evaporation duct becomes lower.

The control parameters for group F are similar to that of group B except that the initial air–sea temperature difference is −2 K (i.e., warmer SST), and accordingly both the sensible and latent heat fluxes are positive. With −*Z*_{i}/*L* ≫ 10 (i.e., F0; Table 3), group F is much more convective than groups D and E. The horizontal grid spacing (accordingly domain size) is doubled (i.e., Δ_{h} = 6 m) and model top is higher (1200 m; see Table 1) for group F to accommodate presumably a deeper BL. A relatively deep well-mixed layer (~between 30 and 600 m) is evident in the wind speed, potential temperature, and specific humidity profiles (Fig. 12). Similar to groups D and E, the wind speed increases over WFLG swell (Fig. 12a). In fact, for both WFLG-64 and WFLG-128, the normalized pressure drag is larger than unity, and accordingly, the wind speed maximum exceeds the geostrophic wind speed *U*_{g}. However, unlike a well-defined wind jet over a relatively thin layer in a stable, neutral or weakly unstable BL, the wind maximum is significantly broader and extends to the upper portion of the MBL, presumably due to stronger vertical mixing of momentum. Over WOPS swell, the pressure drag is nearly 8 times of the shear stress, and the wind speed is substantially reduced throughout the depth of the BL. On the other hand, compared to the marked changes in the wind profiles, the influence of swell on the scalar profiles and, therefore, *M* profile, is rather insignificant. For example, the BL depth only exhibit small variation (~5%) from simulation to simulation, and the *M* profiles almost overlap each other in the lowest 500 m. Nevertheless, there is still up to 3 m increase in the evaporation duct height over WFLG swell (Table 3).

In summary, the influence of swell on an unstable BL is qualitatively similar to that on a SBL, but to a much lesser degree. In an unstable BL, both the vertical wind shear and buoyancy contribute to turbulence production. The influence of swell on an unstable BL becomes progressively less pronounced as the buoyancy production becomes increasingly important in a more convective BL.

## 5. Eddy diffusivity and similarity theory

*K*

_{m}, temperature

*K*

_{θ}, and specific humidity

*K*

_{q}are defined as

*K*

_{θ}and

*K*

_{q}are nearly identical for each simulation, suggesting that there is little dissimilarity between the temperature and water vapor transport. There is a sizable difference between the eddy diffusivities for momentum,

*K*

_{m}, and scalars,

*K*

_{h}=

*K*

_{θ}=

*K*

_{q}, and the ratio between the two, Pr

_{t}=

*K*

_{m}/

*K*

_{h}, is often referred to as the turbulent Prandtl number.

Vertical profiles of eddy diffusivity for (a) momentum, (b) temperature, and (c) their ratio derived from the group A simulations. Only the lowest 30 m is shown. The corresponding eddy diffusivity estimated using the total stress (i.e., sum of shear stress and pressure drag) is also included in (a) as color crosses for comparison.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of eddy diffusivity for (a) momentum, (b) temperature, and (c) their ratio derived from the group A simulations. Only the lowest 30 m is shown. The corresponding eddy diffusivity estimated using the total stress (i.e., sum of shear stress and pressure drag) is also included in (a) as color crosses for comparison.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

Vertical profiles of eddy diffusivity for (a) momentum, (b) temperature, and (c) their ratio derived from the group A simulations. Only the lowest 30 m is shown. The corresponding eddy diffusivity estimated using the total stress (i.e., sum of shear stress and pressure drag) is also included in (a) as color crosses for comparison.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 13, but for group D.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 13, but for group D.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 13, but for group D.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

The eddy diffusivity and Prandtl number profiles derived from group A are shown in Fig. 13. For the FLAT and two WFLG swell simulations, *K*_{m} increases with height over the first few meters and decreases with height aloft. Over WFLG swell *K*_{m} is larger near the surface and becomes smaller aloft than over a flat surface, which is consistent with the variations of vertical wind shear and shear production of turbulence discussed in the previous sections. The *K*_{m} value derived from the WOPS swell simulation is significantly larger than the other three and reaches its maximum at a higher level (i.e., ~15 m). Compared to FLAT, the eddy diffusivity becomes substantially smaller in the presence of WFLG swell, implying that the vertical mixing of scalars is markedly weakened by swell. Particularly, for WFLG-128, the eddy diffusivity decreases to nearly zero at the wind maximum level, where the vertical wind shear is nearly zero. Over a flat surface, the derived turbulence Prandtl number is between 0.35 and 0.5 in the first 10 m and increases to 0.7–0.8 aloft. Over WFLG swell, the Prandtl number reduces to ~0.3 in the lowest 20 m (or lowest ~30%–50% of the BL depth), and over WOPS swell, the Prandtl number becomes larger than unity (i.e., between 1 and 3), implying that eddies are more effective in mixing momentum than scalars.

The eddy diffusivity for the unstable group (i.e., group D; Fig. 14) increases with height over the lowest 50 m with values roughly one order of magnitude larger than their counterpart in group A. The eddy diffusivities are markedly smaller over WFLG swell than over a flat surface, suggesting that the vertical transport of momentum and scalars becomes less efficient over WFLG swell. This is consistent with the reduced shear and weakened TKE in the WFLG simulations. Similar to their counterparts in the stable group, the eddy diffusivities are significantly larger near the surface over WOPS swell than over a flat surface. The Prandtl number is smaller in the lowest 10 m and becomes nearly constant with height in the upper portion of the surface layer.

*ζ*=

*z*/

*L*is the vertical distance nondimensionalized by the Obukhov length, and

*ϕ*

_{m}and

*ϕ*

_{h}denote the empirical universal functions for winds and scalars, respectively. For the purpose of comparison, the eddy diffusivities inferred from groups A and D shown in Figs. 13 and 14 are replotted in Figs. 15 and 16 in the nondimensional form along with (10) and (11) using the empirical universal functions given in the appendix.

The nondimensional eddy diffusivities for the (a) momentum and (b) scalar inferred from group A are shown as circles, with the corresponding curves from MOST. The vertical axis is *ζ* = *z*/*L*_{υ}, where *L*_{υ} is the Monin–Obukhov length.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

The nondimensional eddy diffusivities for the (a) momentum and (b) scalar inferred from group A are shown as circles, with the corresponding curves from MOST. The vertical axis is *ζ* = *z*/*L*_{υ}, where *L*_{υ} is the Monin–Obukhov length.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

The nondimensional eddy diffusivities for the (a) momentum and (b) scalar inferred from group A are shown as circles, with the corresponding curves from MOST. The vertical axis is *ζ* = *z*/*L*_{υ}, where *L*_{υ} is the Monin–Obukhov length.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 15, but for group D.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 15, but for group D.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

As in Fig. 15, but for group D.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0098.1

For group A, the simulated eddy diffusivities for the momentum and scalars over a flat surface show reasonable agreement with the corresponding MOST predictions. For the three swell simulations, the diffusivities from LES deviate substantially from the MOST curves, implying that the MOST becomes invalid over a swell-dominated sea. Over WFLG swell, the simulated *ζ* rapidly at approximately *ζ* ~ 0.03, while their MOST predicted counterparts slowly increase with *ζ*. Over WOPS swell, the simulated eddy diffusivities are smaller than the MOST predictions for *ζ* < 0.3 and become larger than MOST predictions for *ζ* > 0.3. The deviation from the corresponding MOST curves is likely due to the vertical redistribution of turbulence associated with the wind shear regulated by the swell-induced pressure drag.

For group D, the simulated

## 6. Concluding remarks

The impact of swell on the MBL structure, scalar profiles, and radio wave duct characteristics has been examined based on diagnosis of six groups of LES that explicitly resolve both the surface layer and underlying swell. These simulations, including both stable and unstable boundary layers under low to moderate wind speed conditions, highlight the dramatic influence swell may have on mean wind and scalar profiles in a MBL as well as evaporation ducts. Dynamically, the wind speed in near the surface tends to increase over WFLG swell under the influence of forward-pointing pressure drag, and a low-level jet forms when the pressure drag exceeds the shear stress. On the contrary, the wind speed in a MBL can be substantially weakened over WOPS swell due to the momentum removed by both the shear stress and pressure drag, and the MBL thickens accordingly. These results are in qualitative agreement with previous observational and LES studies.

While swell has little direct impact on scalar profiles, we find that it regulates them through changing vertical turbulence mixing. The pressure drag from WFLG swell tends to accelerate (or decelerate over WOPS swell) the wind speed immediately above the wavy surface, leading to enhancement of the wind shear in a shallow layer above the surface and weakening of wind shear aloft. The shear production of turbulence and therefore the turbulence intensity change accordingly. Finally, the swell-induced vertical redistribution of turbulence, evidenced in the eddy diffusivity profiles, regulates the vertical mixing of scalars, resulting in changes in scalar profiles. For example, when a stable BL moving over WFLG swell, the wind shear and turbulence are enhanced immediately above the surface and weakened aloft. Stronger turbulence above the surface leads to stronger vertical mixing and accordingly reduced vertical scalar gradients (i.e., in terms of magnitude). Aloft, weakened turbulence helps maintaining larger temperature or specific humidity gradients. The depth of the layer with enhanced winds is dependent on the both BL and swell characteristics. The weakening of turbulence aloft may significantly reduce the stable BL depth. Over WOPS swell, the pressure drag shares the same sign with the shear stress and the wind speed is reduced throughout a large fraction of the BL. As a result, the shear layer expands substantially, shear production of turbulence becomes more significant, and the BL becomes deeper. The scalar profiles qualitatively resemble those in a weakly convective BL with large gradients in a shallow layer above the surface and reduced gradients in a relatively deep and less stratified layer above. In addition, swell may impact on scalar profiles through changing surface stress and fluxes. For example, the sensible heat flux at the air–sea interface is proportional to the surface wind speed and air–sea temperature difference [see (4)], both of which are subject to change over swell. The Obukhov length and scalar gradients near the surface changes accordingly. However, it is worth noting that the surface fluxes are derived from bulk formula (4) with exchange coefficients computed using constant rough lengths and flux–profile relations from MOST.

The significance of swell impact on a BL and ED appears to vary substantially with the BL stability, BL wind speed, and the swell characteristics. In general, the influence of swell on a stable BL is more dramatic than on an unstable BL. This can be attributed to turbulence generation processes. In a stable BL, stratification tends to suppress turbulence, and shear production is the sole turbulence generation source. Therefore, any change in the wind shear induced by swell manifests itself in the turbulence intensity profile and eventually plays a significant role in vertical turbulence mixing of scalars. In an unstable BL, turbulence can be produced by both shear and buoyancy. In general, the more unstable (convective) the BL is, the less significant the shear production of turbulence plays in modulating the scalar profiles. While the change in wind shear due to swell still has some impact on turbulence intensity and mixing, this impact is much less pronounced than in a SBL. The relative importance of swell effect also varies with the wind speed in the boundary layer. A useful parameter for assessing the relative importance of swell in shaping the BL wind profile is the stress-normalized pressure drag. Over fast propagating WFLG swell, the pressure drag tends to decrease with increasing wind speed and the shear stress approximately increases with the wind speed squared. Therefore, the swell impact appears to be more significant under light winds than strong winds. Finally, the influence from swell on BL structure varies with swell characteristics, such as swell length, slope, and swell–wind alignment angle. The stress-normalized pressure drag varies with these swell parameters substantially. As demonstrated in this study, with forward-pointing pressure drag, WFLG swell tends to boost the low-level winds and decrease the BL height, while the WOPS swell virtually does the opposite.

The Monin–Obukhov similarity theory is still widely used for the prediction of evaporation ducts over ocean where it has never been systematically tested. Its validity over a nonstationary surface such as swell-dominated seas has been questioned for years. Our simulations suggest that the mean winds and scalar profiles over a swell-dominated sea may substantially deviate from classical similarity theory predictions. The deviations from MOST predictions appear to be more dramatic for a stable or a nearly neutral BL over swell and less so for a convective BL. Finally, it is noteworthy that this finding does not necessarily contradict those from Sullivan et al. (2018), which found that the wind and scalar profiles follow the log-linear distribution reasonably well above the wave boundary layer. For the simulations of SBL over WFLG swell examined in this study, the “wave boundary layer” is virtually as deep as the SBL itself, due to the shallowness of the SBL and the use of relatively long monochromatic swell.

## Acknowledgments

This research is supported by the Chief of Naval Research through the NRL Base Program, PE 0601153N. Computational resources were supported by a grant of HPC time from the Department of Defense Major Shared Resource Centers. The author would like to thank Dr. Peter Sullivan for providing his large-eddy simulation code, and Drs. Shouping Wang and Qing Wang for helpful discussions.

## APPENDIX

### Nondimensional Eddy Viscosity

*φ*

_{M}and

*φ*

_{H}are universal functions usually derived empirically from field observations,

*κ*= 0.4 is the von Kármán constant,

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