1. Introduction
Convective boundary layers (CBLs) are frequently observed during daytime as Earth’s surface is warmed by solar radiation (Stull 1988). Due to their frequent occurrence, the fundamental understanding of CBLs is highly relevant to agriculture, architectural design, aviation, climate modeling, weather prediction, and wind energy applications, to name a few. The modern scientific literature on CBLs goes back over 100 years. Initially, the focus was on low-altitude measurements, and with the introduction of more advanced measurement techniques, the focus gradually shifted upward. However, only after the introduction of large-eddy simulations (LES) in the early 1970s, it has become widely accepted that thermodynamic indicators are most suitable to identify the different CBL regions (LeMone et al. 2019). However, obtaining analytical profiles that describe the wind and potential temperature flux in the entire CBL has remained challenging due to the different flow physics in the various CBL regions.
The CBL can be subdivided into three layers (excluding the roughness sublayer), i.e., the surface layer, the mixed layer, and the entrainment zone (see Fig. 1). The surface layer is characterized by a superadiabatic potential temperature gradient and a strong wind shear, which is usually described by the Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954). According to the MOST the nondimensional wind speed and potential temperature gradient profiles are universal functions of the dimensionless height z/L, where z is the height above the surface and L is the surface Obukhov length (Obukhov 1946; Monin and Obukhov 1954). Many studies have pointed out that the MOST does not explain all important surface-layer statistics under convective conditions (Panofsky et al. 1977; Khanna and Brasseur 1997; Johansson et al. 2001; McNaughton et al. 2007; Salesky and Anderson 2020; Cheng et al. 2021) or very stable conditions (Mahrt 1998; Cheng et al. 2005). In particular, the normalized wind gradient
Profiles of the potential temperature Θ, wind speed Umag, and potential temperature flux q in the CBL. The vertical lines with double arrows indicate different length scales in the CBL, namely, from left to right, the Obukhov length L, the lowest height where the potential temperature flux first reaches zero, h1, the inversion-layer height at which the potential temperature flux reaches its minimum value, zi, and the height where the potential temperature flux first recovers zero, h2. The background color indicates the magnitude of the potential temperature flux q(z) for case 2; see Table 1.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0159.1
The mixed layer is characterized by intense vertical mixing caused by warm air thermals rising from the ground. Within the mixed layer, the magnitude of the mean velocity is much larger than the variations in the mean velocity. Thus, for applications where the mean velocity gradient is unimportant, the wind speed and potential temperature can be regarded as uniform (Kaimal et al. 1976; Salesky et al. 2017). This insight is incorporated in various CBL models (Lilly 1968; Deardorff 1973; Stull 1976; Deardorff 1979; Tennekes and Driedonks 1981; Garratt et al. 1982). The entrainment zone is characterized by entrainment of air from the free atmosphere. Deardorff et al. (1980) found in laboratory experiments that the ratio of the entrainment zone thickness to the depth of the mixing layer decreases asymptotically with increasing Richardson number Ri as follows (h2 − h1)/h1 = 0.21 + 1.31/Ri (see Fig. 1 for the definitions of h1 and h2). This ratio is essential for developing entrainment models and has been studied extensively (Lilly 1968; Sullivan et al. 1998; Zilitinkevich et al. 2012; Haghshenas and Mellado 2019). The potential temperature flux profile decreases linearly with height and becomes negative in the entrainment zone. The entrainment flux ratio Πm, which is defined as the ratio of the potential temperature flux at the inversion-layer height to its value at the ground, turns out to be nearly constant, i.e., Πm ≈ −0.2 (Stull 1976; Sorbjan 1996; Conzemius and Fedorovich 2006; Sun and Wang 2008; LeMone et al. 2019). Note that the inversion layer is the upper region of the entrainment zone in which the potential temperature flux increases steeply from its minimum value at z = zi to zero at z = h2 (see Fig. 1).
The geostrophic wind (Ug, Vg) and the friction velocity
Recently, Tong and Ding (2020) analytically derived the convective logarithmic friction law from first principles. They identified three scaling layers for the CBL with −zi/L ≫ 1: the outer layer, the inner-outer layer, and the inner-inner layer. The characteristic length scales for these three layers are the inversion-layer height zi, the Obukhov length L, and the roughness length z0, respectively. The mixed-layer mean velocity scale Um and the geostrophic wind component Vg are the characteristic streamwise and spanwise velocity scales in the outer layer. The difference between the horizontally and temporally averaged velocity U(z) and Um is the mixed-layer velocity-defect law, which has a velocity scale of
Various time-dependent models have been developed to explicitly account for entrainment processes at the top of CBLs (Troen and Mahrt 1986; Noh et al. 2003; Hong et al. 2006). For example, the countergradient transport method (Holtslag and Moeng 1991) and the eddy-diffusivity mass-flux approach (Siebesma et al. 2007; Li et al. 2021) are widely used in coarse-resolution climate models. In general, the potential temperature is time dependent (Lilly 1968) and the entrainment velocity can affect the mean wind speed in the mixed layer (Tong and Ding 2020). However, the velocity and potential temperature flux profiles are quasi stationary, and therefore, similarity theory can be employed to obtain analytical expressions for these profiles shapes (Zilitinkevich and Deardorff 1974; Arya 1975; Zilitinkevich et al. 1992).
In this study, we focus on the derivation of analytical expressions for the mean velocity and potential temperature flux profiles in cloud-free CBLs. We use a perturbation method approach to construct an analytical expression for the normalized potential temperature flux profile as a function of height, taking into account the characteristics of both the surface layer and the capping inversion layer. The depth of the entrainment zone is connected to the convective logarithmic friction law to obtain analytical expressions for the velocity profile in the mixed layer and the entrainment zone. As remarked previously, the surface layer is still described by the MOST.
The organization of the paper is as follows. In section 2 we obtain analytical expression for the potential temperature flux and wind profiles. In section 3 we validate the proposed profiles against LES. The conclusions are given in section 4.
2. Theory
a. Potential temperature flux profile
We note that the perturbation method approach to model the potential temperature flux profile was recently introduced by Liu et al. (2021b) for conventionally neutral atmospheric boundary layers where the surface potential temperature flux is always zero. However, it should be noted that in the CBLs under consideration, the surface is heated and thermal plumes are generated at the ground, resulting in significantly different turbulence generation mechanisms. The applicability of the perturbation method approach to model the strong inversion layer relies on its ability to capture the strong gradients in the inversion layer. Here we used the second-order ODE defined by Eq. (6) to model the potential temperature flux profile as our a posteriori tests confirm that this is sufficient to capture the inversion layer accurately. Higher-order terms could be incorporated, but this is not considered here to keep the obtained profiles relatively simple. An important observation is that the perturbation method approach is consistent with the finding of Garcia and Mellado (2014). They showed that the vertical structure of the entrainment zone is best described by two overlapping sublayers characterized by different length scales, namely, the mean penetration depth of an overshooting thermal for the upper sublayer and the thickness of the CBL for the lower sublayer. Similarly, the second-order ODE, i.e., Eq. (6), indicates that there are two distinct length scales for the description of the entrainment zone (Fig. 1): one is the upper sublayer with zi ≤ z ≤ h2, where the gradient of potential temperature flux is proportional to −(qwΠm)/(2ϵh2), and the other is the lower sublayer with h1 ≤ z ≤ zi, where the gradient of potential temperature flux is proportional to (qwΠm)/zi. Since ϵ ≪ 1 the potential temperature flux varies more steeply in the upper sublayer than in the lower sublayer.
b. Wind profile
Summary of all simulated cases. Here Γ = ∂Θ/∂z is the vertical derivative of the mean potential temperature in the free atmosphere, qw is the surface potential temperature flux, z0 is the roughness length,
3. Numerical validation
a. Numerical method and computational setup
We use LES to simulate the CBL flow over an infinite flat surface with homogeneous roughness. We integrate the spatially filtered Navier–Stokes equations and the filtered transport equation for the potential temperature (Albertson 1996; Albertson and Parlange 1999; Gadde et al. 2021; Liu et al. 2021a,b; Liu and Stevens 2021). Molecular viscosity is neglected as the Reynolds number in the atmospheric boundary layer flow is very high, and we use the advanced Lagrangian-averaging scale-dependent model to parameterize the subgrid-scale shear stress and potential temperature flux (Bou-Zeid et al. 2005; Stoll and Porté-Agel 2008). We note that the Lagrangian-averaging scale-dependent model has been extensively validated and widely used in the literature (Bou-Zeid et al. 2005; Stoll and Porté-Agel 2008; Calaf et al. 2010; Wu and Porté-Agel 2011; Zhang et al. 2019; Gadde et al. 2021).
Our code is an updated version of the one used by Albertson and Parlange (1999). The grid points are uniformly distributed, and the computational planes for horizontal and vertical velocities are staggered in the vertical direction. The first vertical velocity grid plane is located at the ground. The first grid point for the horizontal velocity components and the potential temperature is located at half a grid distance above the ground. We use a second-order finite difference method in the vertical direction and a pseudospectral discretization in the horizontal directions. Time integration is performed using the second-order Adams–Bashforth method. The projection method is used to enforce the divergence-free condition. At the top boundary, we impose a constant potential temperature lapse rate, zero vertical velocity, and zero shear stress boundary condition. At the bottom boundary, we employ the classical wall stress and potential temperature flux formulations based on the MOST (Moeng 1984; Bou-Zeid et al. 2005; Stoll and Porté-Agel 2008; Gadde et al. 2021).
We perform 11 LES to verify the validity of the derived wind speed and potential temperature flux profiles for CBLs. The computational domain is 5 km × 5 km × 2 km and the grid resolution is 256 × 256 × 256. Due to large computational expense, only several external parameters are varied in the simulations. The flow is driven by the geostrophic wind of
A summary of all simulated cases is presented in Table 1. Note that the cases in Table 1 are arranged such that the value of −L/z0 increases monotonically. Furthermore, we note that case 2 has been validated against atmospheric observations, and the simulation results obtained using different subgrid-scale models and grid resolutions is very similar (Gadde et al. 2021). To show the simulated results are independent of the computational domain size, we have performed an additional simulation for case 2 in a larger computational domain (12 km × 6 km × 2 km) on a mesh with 600 × 300 × 240 nodes such that the grid spacings are nearly identical. Figure 2 shows the simulated wind speed and potential temperature flux profiles for case 2. The good agreement between the results obtained with different computational domain size confirms that the simulated results are independent of the computational domain size.
The comparison of simulated (a) normalized wind speed
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0159.1
b. Validation of analytical profiles
Figure 3 shows that with increasing height the normalized potential temperature flux profile q/qw first decreases linearly from unity at the surface to a minimum at zi/h2 = 1 – 2ϵ, before it rapidly increases to zero in the inversion layer (1 – 2ϵ ≤ z/h2 ≤ 1). The normalized thickness of the inversion layer, which is parameterized by ϵ, is expected to depend on the Richardson number (Deardorff et al. 1980), potential temperature gradient (Sorbjan 1996), and wind shear (Conzemius and Fedorovich 2006). However, we find that for the parameter range under consideration, the variation in the normalized thickness of the inversion layer is limited (see Table 1). Therefore, we use a fixed representative value ϵ = 0.044 to model the potential temperature flux profile, and the figure confirms that this ensures that the potential temperature flux profile obtained from the model agrees excellently with all available simulation data, which validates the chosen approach. To further confirm the validity of the potential temperature flux profile, we also compare our results in Fig. 3 with previous LES from Mason (1989), Sorbjan (1996), and Abkar and Moin (2017), the direct numerical simulations data by Garcia and Mellado (2014), and the empirical models by Lenschow (1974) and Noh et al. (2003). Clearly, the model predictions agree well with these previous studies.
Vertical profile of normalized potential temperature flux q/qw. The data are shown as follows: circles, LES data (Table 1); up triangle, LES data of Mason (1989); down triangle, LES data of Sorbjan (1996); square, direct numerical simulations data of Garcia and Mellado (2014); diamond, LES data of Abkar and Moin (2017); red line, prediction given by Lenschow (1974); blue line, prediction given by Noh et al. (2003); black line, prediction given by Eq. (7) with ϵ = 0.044 and cΠ = 1.32. The figure shows that the proposed model captures the simulation trends very well.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0159.1
In Fig. 4a we compare the wind speed in the bulk of the mixed layer (0.4 ≤ z/h2 ≤ 0.6) against the normalized Obukhov length −L/z0 with results from Tong and Ding (2020). The figure shows that our simulations convincingly confirm the validity of the convective logarithmic friction law for the wind speed [Eq. (16) with C = 1] over a much wider range of −L/z0 ∈ [3.6 × 102, 0.7 × 105] than previously considered (−L/z0 ∈ [2.5 × 102, 1.5 × 103]). To further confirm the convective logarithmic friction law, Fig. 4b shows the ratio of the predicted and simulated mixed-layer mean velocity scale
The dependence of (a) the normalized mixed-layer mean velocity scale
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0159.1
In Fig. 5 we compare the vertical profile of the mean streamwise velocity U for four typical cases with different surface potential temperature flux and roughness length with the simulation results. The filled symbols are the present LES data, the dashed line is the theoretical prediction given by the MOST, and the solid line is the prediction given by Eq. (20) with ϵ = 0.044. The figure shows that the MOST accurately captures the surface layer’s wind profile (lowest 20% of the boundary layer). However, in the mixed layer, the prediction of the MOST deviates significantly from the LES data. In particular, the discrepancy from the MOST increases as the surface potential temperature flux qw decreases (Figs. 5b,c) or the roughness length z0 increases (Figs. 5a,d). Therefore, MOST is seldom used to specify wind profiles outside of the surface layer. Figure 5 shows that the proposed wind profile given by Eq. (20) accurately captures the velocity profile throughout the entire boundary layer. This excellent agreement confirms the validity of our proposed wind profile of Eq. (20) for atmospheric boundary layers in the range of studied parameters.
Vertical profile of the mean streamwise velocity U. The data are shown as follows: filled circles, LES data (Table 1); dashed line, prediction given by the MOST; solid line, prediction given by Eq. (20) with ϵ = 0.044. The prediction by the MOST is plotted outside the surface-layer region to demonstrate the difference with the new profile.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0159.1
Figure 6 shows the corresponding profiles of the mean spanwise velocity V. The filled symbols are the present LES data and the solid line is the prediction given by Eq. (22) with ϵ = 0.044. Overall, the agreement between the proposed wind profile given by Eq. (22) and the LES data is reasonably good in the entire boundary layer. This agreement confirms the validity of our proposed wind profile of Eq. (22) for CBLs in the range of studied parameters (i.e., −L/z0 ∈ [3.6 × 102, 0.7 × 105]). We note that the figure confirms that the spanwise velocity V is much smaller than the streamwise velocity U. The figure also indicates that the magnitude of the geostrophic wind component |Vg| increases as the surface potential temperature flux qw decreases (Figs. 6b,c) or the roughness length z0 increases (Figs. 6a,d).
Vertical profile of the mean spanwise velocity V. The data are shown as followes: filled circles, LES data (Table 1); solid line: prediction given by Eq. (22) with ϵ = 0.044.
Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0159.1
4. Conclusions
This work uses a perturbation method approach in conjuncture with the convective logarithmic friction law and the Monin–Obukhov similarity theory to develop analytical expressions of the wind and potential temperature flux profiles in convective atmospheric boundary layers. The validity of the proposed wind [given by Eqs. (20) and (22)] and potential temperature flux profiles [given by Eq. (7)] has been confirmed by their excellent agreement with large-eddy simulations results for atmospheric boundary layers in the convective-roll dominant regime with
Acknowledgments.
This work was supported by the Hundred Talents Program of the Chinese Academy of Sciences, the National Natural Science Fund for Excellent Young Scientists Fund Program (Overseas), the National Natural Science Foundation of China Grant (11621202), the Shell–NWO/FOM initiative Computational sciences for energy research of Shell and Chemical Sciences, Earth and Life Sciences, Physical Sciences, Stichting voor Fundamenteel Onderzoek der Materie (FOM) and STW, and an STW VIDI Grant (14868). This work was sponsored by NWO Domain Science for the use of the national computer facilities. We acknowledge PRACE for awarding us access to Irene at Très Grand Centre de Calcul du CEA (TGCC) under PRACE Project 2019215098, and the advanced computing resources provided by the Supercomputing Center of the USTC.
Data availability statement.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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