1. Introduction
One of the primary sources of uncertainty in mesoscale numerical weather prediction (NWP) models is the representation of thermodynamic processes in the planetary boundary layer (PBL) (Cohen et al. 2015; NielsenGammon et al. 2010). These processes are strongly influenced by the mechanical and thermal mixing induced by Earth’s surface, which is associated with turbulent eddies. The spatiotemporal scales of such eddies cannot be explicitly resolved at the grid scales and time steps typical of NWP models at the mesoscale (Stull 1988). For this reason, PBL parameterization schemes in the framework on the Reynoldsaveraged Navier–Stokes (RANS) equations are employed at the typical resolution of mesoscale models, ranging from hundreds of meters to some kilometers, to parameterize the vertical turbulent flux of momentum, heat, and moisture.
PBL schemes can be divided into two main categories (Zhang et al. 2020): the eddydiffusivity massflux (EDMF) approach and the traditional eddydiffusivity (Ktheory) parameterizations. The EDMF approach consists in the combination of the Ktheory closure, which parameterizes the turbulent transport by small eddies, with the mass flux component accounting for nonlocal organized eddy fluxes (Angevine et al. 2010; Han et al. 2016; Olson et al. 2019). On the other hand, Ktheory turbulence closures can be classified depending on the order of the RANS equations that are resolved. The oneorder turbulence closures estimate the eddy viscosity/diffusivity (ν_{M}, ν_{H}) based on the vertical wind shear and temperature stratification. One example is the wellknown Yonsei State University scheme (YSU; Hong et al. 2006). The 1.5order closures include a prognostic equation for the turbulent kinetic energy (TKE) and a turbulent mixing length scale (ℓ_{K}) for calculating vertical mixing coefficients (K–ℓ approach). The equation for TKE accounts for the contribution of buoyancy, shear, vertical transport, and dissipation rate (ε). The latter is assumed to be proportional to a dissipation mixing length (ℓ_{ε}), set equal to ℓ_{K} in the simplest 1.5order turbulence closures. However, Bougeault and Lacarrere (1989) (hereafter BouLac) utilized two different length scales, depending on atmospheric stability.
An alternative approach to determine the TKE and the vertical mixing coefficients in 1.5order (or higher) closures is to employ an additional prognostic equation for the dissipation rate, in order to avoid defining the diagnostic length scales. This kind of closure, called K–ε hereafter, has been widely used to reproduce vertical PBL profiles in various conditions (Launder and Spalding 1974; Detering and Etling 1985; Duynkerke 1988; Langland and Liou 1996). Beljaars et al. (1987) compared K–ℓbased and K–εbased schemes and found that the K–ε better preserves the “memory effects” of the PBL, because the prognostic equation of ε, including its vertical transport, takes into account its distribution at the previous time step to calculate its temporal evolution, which is not considered in K–ℓ schemes. Wang (2001, 2002) implemented a K–ε scheme in a tropical cyclone model, and later the model was used for regional climate studies, e.g., for investigating the Asian summer monsoon rainfall (Wang et al. 2003; Souma and Wang 2009) and the eastern Pacific boundary layer clouds (Wang et al. 2004a,b; Xie et al. 2007). More recently, Zhang et al. (2020) incorporated the K–ε version of Wang (2001, 2002) in the Weather Research and Forecasting (WRF) Model (Skamarock et al. 2019). The new scheme was evaluated in the PBL topped by stratocumulus clouds over the southeast Pacific and the southern Great Plains, finding that the K–ε performed similarly to other stateoftheart PBL schemes. However, several studies (Launder and Spalding 1983; Sukoriansky et al. 2005; Lazeroms et al. 2015; van der Laan et al. 2017; Zeng et al. 2020; Zeng and Wang 2020) highlighted the necessity to modify the standard K–ε turbulence closure, since it does not perform well in both convective and stable regimes, especially in flows with strong mean shear. Although Zhang et al. (2020) showed good performance of their K–ε scheme as mentioned above, recent insights have even shown that additional prognostic equations are required, in particular for the potential temperature variance (Mauritsen et al. 2007; Zilitinkevich et al. 2007, 2013).
This work shows the advantage of a new K–ε turbulence closure, with appropriate modifications, to face the aforementioned problems, to improve its capability of reproducing simple idealized cases when implemented in the WRF mesoscale model. Specifically, the standard K–ε closure is modified through (i) the estimation of the vertical profile of the Prandtl number as in Hong et al. (2006), to take into account the difference between eddy diffusivity and eddy viscosity, (ii) an additional correction term in the prognostic equation for the dissipation rate as in Zhang et al. (2020), to make the new closure consistent with the Monin–Obukhov similarity theory (MOST), (iii) the coupling with a prognostic equation for the turbulent potential energy (TPE; proportional to the temperature variance) as in Lazeroms et al. (2015) and Želi et al. (2019), to consider its effect on the turbulent heat flux. The latter, for convective cases, is also compared with a closure employing a nonlocal countergradient term, computed as in Ching et al. (2014). The novel PBL scheme is tested by means of idealized simulations. Idealized simulations include several flat terrain cases, with different thermal stratification in both convective and stable regimes, and a complex terrain case in convective conditions with various wind forcing. The aim is to assess if the employment of additional prognostic equations, considering “memory effects” and turbulent transport of the dissipation rate and temperature variance, is beneficial when reproducing PBL processes. The newly developed PBL scheme is validated against ensembles of largeeddy simulations (LES), taken as reference for each case study, and compared with stateoftheart PBL schemes, at different orders, already implemented in WRF.
The paper is organized as follows: the theory of the newly introduced PBL scheme, along with the novel computational solution is presented in section 2. The setup of the five idealized case studies and the methodology for the calculation of the turbulent fluxes are described in section 3. In section 4 model outputs are compared with LES for each case study, and the performance of the various PBL schemes is quantified through statistical parameters. Finally, in section 5, results are summarized and discussed.
2. The model
The turbulence parameterization scheme presented here is developed in the framework of the RANS equations, in which each variable of the mean flow is decomposed into its mean, representing an ensemble average (uppercase letters), and fluctuating part (lowercase letters). Planetary boundary layer (PBL) parameterizations generally assume horizontal homogeneity, to consider only the vertical derivative of the turbulent fluxes. Then turbulent contribution to the mean flow dynamics is given by the following:

Zonal wind speed:$\frac{\partial U}{\partial t}=\frac{\partial \overline{uw}}{\partial z}.$

Meridional wind speed:$\frac{\partial V}{\partial t}=\frac{\partial \overline{\upsilon w}}{\partial z}.$

Potential temperature:$\frac{\partial \mathrm{\Theta}}{\partial t}=\frac{\partial \overline{w\theta}}{\partial z}.$
The quantities
a. The standard K–ε turbulence closure
b. The correction term for the ε equation
c. The countergradient heat flux
d. The temperature variance equation
e. The numerical solver
Equations for wind speed, potential temperature, and water vapor mixing ratio are solved implicitly, using the tridiagonal matrix algorithm, adding source and sink terms at the surface (as explained later). For the coupled equations of K and ε, a more complex method is needed, since the strong nonlinearities may interact with discretization errors in such a way to destabilize computation (Lew et al. 2001).
f. Initial and boundary and conditions
3. Setup and case studies
In this study the Advanced Research version of the Weather Research and Forecasting (WRF) Model, version 4.1, is used for the numerical simulations (Skamarock et al. 2019). WRF has been successfully applied in several studies for idealized cases for both RANS simulations and LES, both in flat (Zhang et al. 2018; Moeng et al. 2007) and in complex terrain (Schmidli et al. 2011; Wagner et al. 2014). The thirdorder Runge–Kutta method is used for time integration for all the simulations in this study. In LES mode, we use the 1.5order 3DTKE model for the subgrid turbulence parameterization (Deardorff 1980). We take a threedimensional average of 20 different LES, differing in the initial random potential temperature perturbation, with amplitude of 0.1 K and zero mean, applied at the first four vertical layers, necessary to trigger turbulence at the initial time step. For the very stable case study (viz. GABLS) LES are the ones used in Beare et al. (2006) (available at https://gabls.metoffice.com/lem_data.html). LES are considered as our reference for each case analyzed, and are compared with RANS simulations performed with the novel K–ε schemes presented here, which have been implemented in the WRF Model, and with other conventional 1D PBL schemes already implemented in the standard version of the WRF Model. These PBL schemes are the BouLac (Bougeault and Lacarrere 1989), the Mellor–Yamada–Nakanishi–Niino level 2.5 without the massflux component (MYNN2.5; Nakanishi and Niino 2004) and the Yonsei State University (YSU; Hong et al. 2006) schemes. Moreover, the comparison is performed with a K–εbased scheme (E–ε hereafter, Zhang et al. 2020), introduced in WRF 4.3, and implemented for this work in WRF 4.1. Both RANS simulations and LES are performed with a time step of 0.5 s, for a 4h period (with the exception of the GABLS simulations, which cover 9 h). Lateral boundary conditions in both W–E and S–N directions are periodic, allowing to replicate an infinite domain. The results from the different RANS simulations are compared with the LES considering the hourly average values computed on all the time steps between the third and the fourth hour of simulation, when the PBL is well developed. Similarly, simulation outputs are averaged horizontally, in order to compare model results on a single column value for the flat cases (average in both horizontal directions) and on a crossvalley section for the valley cases (average along the southnorth direction). All simulations are performed for a dry atmosphere, with zero humidity both in the air and in the soil. In this study, five different cases (summarized in Table 1) are considered, varying in thermal stratification, surface temperature forcing, orography, and geostrophic wind. In section 4 we will present the comparison, for the five case studies, between each reference LES, the aforementioned conventional PBL schemes, and two different versions of the K–ε closure: the first experiment (K–ε–γ hereafter) assuming a countergradient term dependent only on the surfacelayer features, calculated as in Eq. (12), and the second experiment (K–ε–θ^{2} hereafter), in which the equation for the temperature variance is calculated from Eq. (13) and the countergradient term is computed using Eq. (18). Moreover, in the stable case (called GABLS hereafter), we run the K–ε–θ^{2} closure also removing the additional term in the buoyancy production of the dissipation rate equation [Eq. (10)], in order to evaluate its contribution in improving the reproduction of the PBL in stable regimes (K–ε–θ^{2}–NOA_{ε} hereafter).
Schematic overview of the different case studies.
a. CBL on flat terrain
b. CBL in an idealized valley
c. SBL in flat terrain
This case is based on the simulations of an Arctic SBL [GEWEX Atmospheric Boundary Layer Study (GABLS)] presented in Kosović and Curry (2000), and subsequently used for the intercomparison of LES (Beare et al. 2006) and RANS (Cuxart et al. 2006) simulations, aiming to quantify the reliability of different PBL schemes through the comparison with observational data. The initial potential temperature profile consists of a mixed layer up to 100 m with a potential temperature of 265 K, with an overlying inversion with Γ = 10 K km^{−1}. A surface cooling rate of 0.25 K h^{−1} is applied for 9 h, so a quasiequilibrium state is reached. The geostrophic wind is set to 8 m s^{−1} in the W–E direction, with a Coriolis parameter of 1.39 × 10^{−4} s^{−1}.
For this case study, we take as reference the LES presented in Beare et al. (2006) at 3.125m vertical and horizontal resolution [viz., CORA, CSU, IMUK, LLNL, NCAR, NERSC, and UIB in Beare et al. (2006)]. They are produced by different NWP models and adopt various subgrid turbulence closures. The simulation domain is a box of 400 m × 400 m × 400 m, and simulations outputs are averaged spatially over the horizontal domain and temporally between the eighth and ninth hour of the simulation. RANS simulations instead are run on a 10 km × 10 km horizontal domain; the top of the domain is set at 1 km above ground level, adopting a depth of 5 m for each vertical level.
d. Calculation of turbulent fluxes
4. Results
This section presents the results of the comparison between the idealized RANS simulations with the different PBL schemes and the LES, for the different case studies shown in Table 1. In the following subsections (4a, 4b, and 4c), the presentation of the results for the different case studies is separated between CBL in flat terrain (CBL_F_3,CBL_F_10), CBL in an idealized valley (CBL_V_NOW,CBL_V_W), and SBL in flat terrain (GABLS), respectively.
a. CBL in flat terrain
Figure 2 shows the vertical profiles of wind speed (left) and potential temperature (right) for the CBL_F_3 case, considering the temporal average between the third and the fourth hour of time integration. The profile is typical of a CBL, with a surface layer ∼150 m deep and a PBL height of ∼1500 m. The wind speed follows this pattern, approaching ∼10 m s^{−1} over the PBL, remaining almost constant in the mixed layer and rapidly decreasing to ∼5 m s^{−1} in the first vertical level. All RANS simulations reasonably agree with the LES in terms of potential temperature. In particular, K–ε–θ^{2} outperforms the other schemes within the surface layer, while BouLac is the best in reproducing the capping inversion over the mixed layer. On the other hand, YSU overestimates the PBL height, while MYNN2.5 exhibits a quasiunstable boundary layer instead of a mixed layer. E–ε well performs in terms of gradients, but it underestimates the temperature in the mixed layer, because of a too deep surface layer, and underestimates the height of the PBL. K–ε–γ reasonably reproduces the mixed layer, but it overestimates the absolute value of the potential temperature gradient in the surface layer, and it slightly overestimates the PBL height. Regarding the wind speed profile, YSU and E–ε are the best in reproducing the vertical profile within the surface and the mixed layer, but, as for the potential temperature, the overestimation (underestimation) of the PBL height leads to an underestimation (overestimation) of wind speed in the capping layer, respectively. Despite a good performance in terms of potential temperature, BouLac fails in reproducing the wind profile in the mixed layer, while MYNN2.5 underestimates the wind speed in the capping layer. K–ε–θ^{2}, instead, performs similarly to MYNN2.5 in the mixed layer, but it is the best simulation in reproducing the wind shear in the capping layer. K–ε–γ performs similarly to K–ε–θ^{2}, but again overestimating the PBL height and then underestimating the wind speed in the capping layer.
Vertical profiles of (a) wind speed and (b) potential temperature for the CBL_F_3 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) wind speed and (b) potential temperature for the CBL_F_3 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) wind speed and (b) potential temperature for the CBL_F_3 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
To quantify the ability of the RANS simulations in reproducing wind speed and potential temperature in the CBL, in Fig. 3 we show the rootmeansquare error (RMSE), calculated along the air column, between each RANS simulation and the reference LES. K–ε–γ, K–ε–θ^{2}, MYNN2.5, and YSU show RMSE ∼ 0.3–0.35 m s^{−1} for the wind speed, while in BouLac the RMSE is considerably higher (∼0.6 m s^{−1}). The best results are shown by E–ε, with an error of ∼0.2 m s^{−1}. On the other side, BouLac is the best in reproducing the potential temperature profile (RMSE ∼ 0.1°C), followed by K–ε–θ^{2} and MYNN2.5. The highest errors are shown by K–ε–γ, YSU and E–ε, since the first two overestimate the height of the inversion layer, while the latter cannot reproduce correctly the temperature in the mixed layer.
Rootmeansquare error (RMSE) for each RANS simulation with respect to the reference LES, calculated for an air column up to 2050 m above ground level for the CBL_F_3 case, for (a) wind speed and (b) potential temperature.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Rootmeansquare error (RMSE) for each RANS simulation with respect to the reference LES, calculated for an air column up to 2050 m above ground level for the CBL_F_3 case, for (a) wind speed and (b) potential temperature.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Rootmeansquare error (RMSE) for each RANS simulation with respect to the reference LES, calculated for an air column up to 2050 m above ground level for the CBL_F_3 case, for (a) wind speed and (b) potential temperature.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Figure 4 reports the vertical profiles of the vertical heat flux (left) and of the vertical momentum flux (right).
Vertical profiles of (a) vertical momentum flux and (b) vertical heat flux for the CBL_F_3 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical momentum flux and (b) vertical heat flux for the CBL_F_3 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical momentum flux and (b) vertical heat flux for the CBL_F_3 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Figure 5 shows the vertical profiles of wind speed (left) and potential temperature (right) for the CBL_F_10 case. This case differs from the previous one only in the initial vertical temperature gradient, which is now set to 10 K km^{−1}. Due to this stronger stratification, the PBL height reaches a depth of ∼800 m, with a less unstable surface layer with respect to the previous case. While there are no relevant differences between the RANS simulations in the surface layer, in the mixed layer and in the entrainment zone K–ε–θ^{2} provides the best results regarding the temperature profile, followed by K–ε–γ. On the other hand, YSU and BouLac overestimate, while MYNN2.5 and E–ε underestimate the inversion layer height. In particular, E–ε cannot reproduce the slope of the temperature profile, probably due to a too low value of the countergradient term. Improvements by the K–ε schemes are found even in reproducing the wind speed, with best results again in the capping layer. Again, BouLac shows a too low wind speed in the mixed layer, while MYNN2.5 over the capping layer. RMSEs (Fig. 6) show indeed the lowest values for the two K–ε schemes, even onethird with respect to the other RANS simulations for the potential temperature. They are followed by YSU, while MYNN2.5 and BouLac and E–ε are the worst, since they cannot capture precisely the capping layer height. Even for the turbulent fluxes (Fig. 7), better results are found for K–ε–θ^{2}: while in the surface and mixed layers it is not possible to identify particular differences between the different RANS simulations, at the inversion layer the new turbulence scheme can better reproduce the negative peak of the vertical heat flux (left) and the slope of the vertical momentum flux (right).
Vertical profiles of (a) wind speed and (b) potential temperature for the CBL_F_10 case. The dashed black line refers to the ensemble of LES simulations, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) wind speed and (b) potential temperature for the CBL_F_10 case. The dashed black line refers to the ensemble of LES simulations, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) wind speed and (b) potential temperature for the CBL_F_10 case. The dashed black line refers to the ensemble of LES simulations, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Rootmeansquare error (RMSE) for each RANS simulation with respect to the reference LES, calculated for an air column up to 1000 m for the CBL_F_10 case, for (a) wind speed and (b) potential temperature.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Rootmeansquare error (RMSE) for each RANS simulation with respect to the reference LES, calculated for an air column up to 1000 m for the CBL_F_10 case, for (a) wind speed and (b) potential temperature.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Rootmeansquare error (RMSE) for each RANS simulation with respect to the reference LES, calculated for an air column up to 1000 m for the CBL_F_10 case, for (a) wind speed and (b) potential temperature.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical momentum flux and (b) vertical heat flux for the CBL_F_10 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical momentum flux and (b) vertical heat flux for the CBL_F_10 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical momentum flux and (b) vertical heat flux for the CBL_F_10 case. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
b. CBL in an idealized valley
In this subsection, the results from the simulations for the idealized valley are presented. Figure 8 shows the crossvalley section of zonal wind (left), potential temperature (middle), and meridional wind (right) averaged along the N–S direction and from the third to the fourth hour of simulation for the CBL_V_W case study for the reference ensemble of LES (CBL_V_NOW shows similar patterns for U and Θ). A crossvalley circulation is well distinguishable from the zonal wind speed panel, with two crossvalley circulation cells on top of each other, similar to those identified in Wagner et al. (2014). Upslope winds are weaker close to the valley floor, and they reach a maximum value of ∼4 m s^{−1} close to the ridge. A return flow toward the center of the valley is evident between 1500 and 2500 m: warmer air is advected from the ridge top to the center of the valley, despite the presence of an underlying smaller thermal convective cell, in analogy with what found in Serafin and Zardi (2010). Because of this return flow, the potential temperature profile exhibits a double mixed layer, one over the surface layer, and the second at the level of the returnflow layer. The presence of upslope circulations along the ridges increases the wind shear, decreasing the meridional wind speed through increased turbulence production. Indeed, the vertical profile of meridional wind speed (right panel of Fig. 8) is not constant along the valley slope, but it is influenced by the branch of the upslope circulation pointing toward the center of the valley. As a consequence, above the ridge level, the meridional wind speed is lower.
Zonal section of (left) zonal wind speed, (center) potential temperature, and (right) meridional wind speed, for the ensemble of LES of the CBL_V_W case study. Vertical black lines refer to the position of vertical profiles shown in Figs. 9 and 12 for the lower point, and in Figs. 10 and 13 for the upper point along the slope.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Zonal section of (left) zonal wind speed, (center) potential temperature, and (right) meridional wind speed, for the ensemble of LES of the CBL_V_W case study. Vertical black lines refer to the position of vertical profiles shown in Figs. 9 and 12 for the lower point, and in Figs. 10 and 13 for the upper point along the slope.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Zonal section of (left) zonal wind speed, (center) potential temperature, and (right) meridional wind speed, for the ensemble of LES of the CBL_V_W case study. Vertical black lines refer to the position of vertical profiles shown in Figs. 9 and 12 for the lower point, and in Figs. 10 and 13 for the upper point along the slope.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Figure 9 shows the vertical profiles of zonal wind speed (left) and potential temperature (right) for a point along the eastern slope situated at 267 m above the valley floor for the CBL_V_NOW case study. The zonal wind speed presents four different peaks, with the lowest and the highest ones more intense than the other two in absolute value, representing the two convective cells described before (negative values represent air moving from the ridge to the valley center). All RANS simulations are able to capture the double circulation along the vertical, but with some errors. In particular, YSU overestimates the height of the three upper peaks, BouLac and MYNN2.5 overestimates the third peak, E–ε uderestimates the first peak and overestimates the third, while the two K–ε underestimate the second and the fourth peak, but they are the best in reproducing the third peak.
Vertical profiles of (a) zonal wind speed and (b) potential temperature for the CBL_V_NOW case for a point situated at 267 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed and (b) potential temperature for the CBL_V_NOW case for a point situated at 267 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed and (b) potential temperature for the CBL_V_NOW case for a point situated at 267 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Regarding the potential temperature profile, the two K–ε schemes better describe the surface layer and the lower mixed layer and inversion, while the others cannot capture correctly the height of the lower inversion (BouLac, E–ε, and MYNN2.5) or underestimate its gradient (YSU). All RANS simulations can capture the higher mixed layer above the ridge level, with a better performance of the two K–ε, both in terms of depth and absolute value.
Figure 10 shows the vertical profiles of zonal wind speed (left) and potential temperature (right) for a point along the eastern slope situated at 1232 m above the valley floor, i.e., where only the upper cell is present, for the CBL_V_NOW case study. For this reason, zonal wind vertical profiles display just the peak close to the surface and the peak of the return wind above the ridge level (around 800 m above ground level). All RANS simulations capture the cell circulation, but E–ε overestimates the intensity of the first peak, and all PBL schemes overestimate the height of the second peak (in particular YSU). However, the two K–ε schemes get closer to the LES in representing the peak, and can better capture the decrease of wind speed with height (especially from 1200 to 1600 m above ground level). The two K–ε display better results considering especially the vertical profile of potential temperature (right panel of Fig. 10). In particular, K–ε–θ^{2} shows a better agreement for the air column with respect to the other RANS simulations. BouLac underestimates the potential temperature along all the vertical profile, while E–ε fails to capture adequately the structure of the first mixed layer. Finally, MYNN2.5 often displays an overestimation, while YSU reproduce the presence of both the mixed layers.
Vertical profiles of (a) zonal wind speed and (b) potential temperature for the CBL_V_NOW case for a point situated at 1232 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed and (b) potential temperature for the CBL_V_NOW case for a point situated at 1232 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed and (b) potential temperature for the CBL_V_NOW case for a point situated at 1232 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
To quantify the ability of each RANS simulation to reproduce the thermal circulation of the CBL_V_NOW case study, in Fig. 11 we show the RMSE, calculated along the first 170 vertical levels (∼1800 m AGL), of zonal wind speed (left) and potential temperature (right) along the eastern slope of the valley. In general, the highest errors for the zonal wind are located in the parts of the slope close to the valley floor and close to the ridge, while they are very low where the wind tends to zero, i.e., above the valley floor and above the ridge. K–ε–θ^{2} presents the lowest RMSE, always between 0.2 and 0.4 m s^{−1} in the central part of the slope. YSU is the worst in reproducing the wind profile in the highest part of the slope (RMSE up to 1 m s^{−1}), while E–ε presents the highest RMSE above the valley floor (up to 1 m s^{−1}).
RMSEs of (a) zonal wind speed and (b) potential temperature with respect to the ensemble of LES, calculated as the average of the first 170 vertical levels for each point from the valley floor to the eastern ridge, for the CBL_V_NOW case for each RANS simulation. The background gray area represents the height above the valley floor along the W–E direction.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
RMSEs of (a) zonal wind speed and (b) potential temperature with respect to the ensemble of LES, calculated as the average of the first 170 vertical levels for each point from the valley floor to the eastern ridge, for the CBL_V_NOW case for each RANS simulation. The background gray area represents the height above the valley floor along the W–E direction.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
RMSEs of (a) zonal wind speed and (b) potential temperature with respect to the ensemble of LES, calculated as the average of the first 170 vertical levels for each point from the valley floor to the eastern ridge, for the CBL_V_NOW case for each RANS simulation. The background gray area represents the height above the valley floor along the W–E direction.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Regarding the RMSE for potential temperature, again K–ε simulations show the best agreement with respect to the LES, especially in the lower part of the valley, followed by MYNN2.5 and YSU. BouLac is constantly the worst in terms of potential temperature, due to its constant underestimation over all the vertical column, followed by E–ε. While K–ε simulations maintain a constant value of the RMSE along the slope, YSU and MYNN2.5 errors increase approaching the valley floor. For all RANS simulations, the highest RMSEs are found above the valley floor and the ridge.
Figures 12 and 13 show the vertical profile of zonal wind speed (left), potential temperature (center), and meridional wind speed (right) for the CBL_V_W case, for two points situated on the eastern slope at 267 and 1232 m above the valley floor, respectively. CBL_V_W differs from CBL_V_NOW for an imposed meridional geostrophic wind of 10 m s^{−1}, with the aim of simulating an alongvalley wind and its interaction with the crossvalley thermal circulation.
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the CBL_V_W case for a point situated at 267 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the CBL_V_W case for a point situated at 267 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the CBL_V_W case for a point situated at 267 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the CBL_V_W case for a point situated at 1232 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the CBL_V_W case for a point situated at 1232 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the CBL_V_W case for a point situated at 1232 m above the valley floor on the eastern slope. The dashed black line refers to the ensemble of LES, while colored lines refer to the different RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Regarding the crossvalley wind and the potential temperature profiles, no substantial differences can be noticed with respect to the CBL_V_NOW case. K–ε–θ^{2} is the most accurate in the simulation of both wind speed and potential temperature, especially in terms of potential temperature for the lower point (267 m above the valley floor) and of wind speed for the upper point (1232 m above the valley floor). The presence of the meridional wind causes stronger differences between the results of the two K–ε with respect to the CBL_V_NOW case. In fact, K–ε–θ^{2} is more precise than K–ε–γ, since it better reproduces the upslope wind intensity and the potential temperature in the surface layer at both heights. Most likely, the prognostic calculation of the countergradient flux in K–ε–θ^{2} becomes more efficient, with respect to a diagnostic value (assumed by K–ε–γ), with the increasing complexity of PBL dynamics. Indeed, the largest improvements take place for the potential temperature in the lowest levels where, in unstable conditions, the temperature variance is larger with respect to the upper levels, as shown initially by Willis and Deardorff (1974). The vertical profile of the meridional wind is more complex than in the flat case (left panel in Fig. 2), especially for the point at 267 m above the valley floor. This is the effect of the cross valley circulation: the thermal crossvalley circulation increases the wind shear and consequently increases the turbulence production, resulting in a decrease of wind speed in correspondence to the maximum wind shear. All RANS simulations, except for YSU, can capture the effect of the crossvalley thermal circulation on the alongvalley meridional wind, and also the different intensity between the lower and upper points. The inefficiency of YSU in representing the vertical profiles of the alongvalley wind is probably linked to the lack of a prognostic equation for TKE of this turbulence parameterization (developed for flat uniform terrains), since more complex parameterizations become more efficient with increasing complexity (Chrobok et al. 1992). K–ε–θ^{2} is the most efficient in capturing the height of the local minima and maxima of the along valley wind for the lower point, despite the overestimation (underestimation) of the maximum (minimum) at 900 m (1300 m). The same occurs for the higher point, even if the meridional wind speed is underestimated between 500 and 1000 m above the valley floor level. On the other hand, K–ε–θ^{2} performs better with respect to the other RANS simulations (which overestimate the height of the inversion layer) in reproducing the meridional wind speed from 1000 to 1600 m, i.e., at the height of the inversion layer, for the upper point. Even for CBL_V_W, K–ε–θ^{2} performs better with respect to K–ε–γ, especially in terms of the intensity of the various peaks.
Figure 14 displays the RMSE, calculated along the first 170 vertical levels, of the zonal wind speed (left), potential temperature (center), and meridional wind speed (right) for each point on the eastern slope of the valley. As in the CBL_V_NOW case, the two K–ε simulations perform better in the reproduction of the cross valley wind, especially for the points close to the ridge and to the valley floor. In this case, the difference between K–ε–θ^{2} and K–ε–γ is higher, with the first that maintains good results close to the valley floor, where the performance of the other RANS simulations decreases significantly.
RMSEs of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature with respect to the ensemble of LES, calculated for the first 170 vertical levels for each point from the valley floor to the eastern ridge, for the CBL_V_W for each RANS simulation. The background gray area represents the height above the valley floor along the W–E direction.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
RMSEs of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature with respect to the ensemble of LES, calculated for the first 170 vertical levels for each point from the valley floor to the eastern ridge, for the CBL_V_W for each RANS simulation. The background gray area represents the height above the valley floor along the W–E direction.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
RMSEs of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature with respect to the ensemble of LES, calculated for the first 170 vertical levels for each point from the valley floor to the eastern ridge, for the CBL_V_W for each RANS simulation. The background gray area represents the height above the valley floor along the W–E direction.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
K–ε–θ^{2} displays good results also for the potential temperature profile (middle panel). As for the CBL_V_NOW case, BouLac and E–ε present the worst results, since they always underestimate the potential temperature in the air column. In this case, MYNN2.5 performs similarly to K–ε–θ^{2}, while K–ε–γ fails to satisfactorily reproduce the potential temperature in the points close to the valley floor.
Regarding the alongvalley wind, K–ε–θ^{2} reveals to be the best for the points close to the ridge, where also K–ε–γ shows lower RMSEs with respect to the other simulations. Approaching the valley floor, all RANS simulations present similar errors, while above the valley floor, where the influence of the sidewalls is lower, YSU performs better than the others.
c. SBL in flat terrain
Figure 15 displays the vertical profiles of zonal wind speed (left), potential temperature (center), and meridional wind speed (right) for the GABLS case study, averaged between the eighth and ninth hour of simulation. LES, performed with different atmospheric models, show a boundary layer height between 150 and 200 m, as stated in Beare et al. (2006), and predict a lowlevel jet (in the S–N direction) forced by the Coriolis term, as well as a peak in the zonal wind speed corresponding to the PBL height. The best RANS simulation in reproducing both wind speed and potential temperature is again K–ε–θ^{2}, which can correctly capture the height of the boundary layer, while K–ε–γ (where γ is zero in stable conditions) slightly overestimates the PBL height. Also YSU overestimates the PBL height, and, in addition, it underestimates the potential temperature gradient in the inversion layer, resulting in smoother peaks for both U and V at the top of the PBL. MYNN2.5 performs similarly to K–ε–θ^{2}, but it underestimates the magnitude of the lowlevel jet. E–ε performs similarly to YSU for all variables, slightly overestimating the PBL height and underestimating the strength of the inversion layer. On the other hand, BouLac fails in reproducing all the vertical profiles, since it does not capture the correct shape of the potential temperature profile and, as a consequence, wrongly estimates the PBL height (∼120 m) and overestimates the inversion layer height (∼360 m, while it should be ∼200 m). The lack of BouLac in correctly representing the vertical profiles in this specific stable regime is probably due to the wrong calculation of the correct length scale, which strongly depends on the atmospheric stability (see Bougeault and Lacarrere 1989 for further details). Similarly, K–ε–θ^{2}–NOA_{ε}, which does not include the additional term in the dissipation rate equation [Eq. (10)], largely overestimates the PBL height (∼250 m), and overestimates also the height of the inversion layer, with even larger errors than in BouLac for the wind, while for potential temperature it performs similarly to the other RANS. This overestimation underlines the importance of the additional term A_{ε} for the dissipation rate equation, which depends linearly on N and on ε itself, increases the dissipation rate where N is higher (i.e., between 100 and 250 m), hence reducing the PBL height and showing a better agreement with the LES. Indeed, as shown in Fig. 16 (left panel), the TKE of K–ε–θ^{2}–NOA_{ε} is highly overestimated, while the TKE of K–ε–θ^{2} and K–ε–γ lays always in the range of the different LES. The reduction of the TKE is due to the increase of the dissipation rate in the higher levels (center panel). Since the eddy viscosity and diffusivity are inversely proportional to the dissipation rate, an increase in ε corresponds to a decrease of ν_{M} and ν_{H}, and hence in a decrease of TKE, heat and momentum flux. These results confirm that the inclusion of A_{ε} allows a better representation of the turbulent variables, even considering the temperature variance (right panel of Fig. 16). The
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the GABLS case. The horizontal lines on the left of (a) show the diagnostic boundary layer height for the RANS simulations and LES. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the GABLS case. The horizontal lines on the left of (a) show the diagnostic boundary layer height for the RANS simulations and LES. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) zonal wind speed, (b) meridional wind speed, and (c) potential temperature for the GABLS case. The horizontal lines on the left of (a) show the diagnostic boundary layer height for the RANS simulations and LES. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) TKE, (b) dissipation rate, and (c) temperature variance for the GABLS case. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) TKE, (b) dissipation rate, and (c) temperature variance for the GABLS case. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) TKE, (b) dissipation rate, and (c) temperature variance for the GABLS case. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Figure 17 displays the wind odographs from the different simulations. All the simulations, apart from K–ε–θ^{2}–NOA_{ε} (which fails throughout the whole vertical air column), well capture the intensity and the direction of the wind speed in the surface layer. At the peak of the lowlevel jet (where V ∼ 3 m s^{−1} and U ∼ 6.5 m s^{−1}), MYNN 2.5 underestimates, while BouLac and K–ε–θ^{2} slightly overestimate the lowlevel jet intensity. Above the lowlevel jet peak, all the RANS simulations behave well returning to geostrophic conditions, while the two failing in capturing the PBL height (BouLac and K–ε–θ^{2}–NOA_{ε}), reach the geostrophic conditions at higher levels.
Hodographs of the mean velocity vector for the GABLS case. The series of dashed black lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Hodographs of the mean velocity vector for the GABLS case. The series of dashed black lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Hodographs of the mean velocity vector for the GABLS case. The series of dashed black lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Figure 18 shows the vertical profiles of zonal momentum flux (left), meridional momentum flux (center) and heat flux (right). While BouLac and K–ε–θ^{2}–NOA_{ε} largely overestimate the absolute value of the momentum fluxes, due to an overestimation of the turbulent production (Fig. 16a), the other RANS simulations reasonably reproduce the vertical profile of the fluxes. In particular, the surface heat flux
Vertical profiles of (a) vertical zonal momentum flux, (b) vertical meridional momentum flux(c) and vertical heat flux for the GABLS case. The horizontal lines on the left of (a) show the diagnostic boundary layer height for the RANS simulations and the LES. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical zonal momentum flux, (b) vertical meridional momentum flux(c) and vertical heat flux for the GABLS case. The horizontal lines on the left of (a) show the diagnostic boundary layer height for the RANS simulations and the LES. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
Vertical profiles of (a) vertical zonal momentum flux, (b) vertical meridional momentum flux(c) and vertical heat flux for the GABLS case. The horizontal lines on the left of (a) show the diagnostic boundary layer height for the RANS simulations and the LES. The series of dashed gray lines refers to the different LES, while color lines refer to the RANS simulations.
Citation: Monthly Weather Review 150, 8; 10.1175/MWRD210299.1
5. Summary and conclusions
As the demand for more accurate NWP models increases, especially for complex and heterogeneous terrain, the development of more precise and local turbulence closures is required. To this end, a novel onedimensional 1.5order parameterization scheme has been developed, based on the coupled equations for TKE and ε, and included in the WRF Model, with the aim to implement a RANS turbulence closure independent from any length scale. The standard K–ε turbulence scheme has been improved by calculating the turbulent Prandtl number (similarly to Hong et al. 2006) the prognostic equation for temperature variance (Lazeroms et al. 2016), and including an additional term to better reproduce the dissipation rate in stable regimes (Zeng et al. 2020).
Since NWP models are adopted in all areas worldwide, particular attention is paid to the numerical method adopted to solve this set of equations, to obtain the most stable numerical integration that can work at time steps suitable for forecasting applications. With this goal in mind, an analytical solution for the TKE and ε equations is derived. In this way, the numerical solution is stable and can work with large time steps.
The new turbulence closure is tested in various idealized case studies, under different atmospheric stability and terrain complexity conditions. Tests include the convective boundary layer in flat terrain, the convective thermal circulation induced by a valley, and the wellknown GABLS case for the very stable boundary layer. For each test case, our K–ε parameterizations (in two different forms, differing in the parameterization adopted for the countergradient term) have been tested against an ensemble of LES, differing in the initial temperature perturbation, and against stateoftheart RANS turbulence schemes at the first order (YSU), or depending on various diagnostic length scales (BouLac and MYNN2.5).
Results show that in general the novel K–ε scheme always outperforms the other parameterizations, for both wind speed and potential temperature, in all the cases considered in this work. In particular, the largest improvements are found in connection with inversion layers, where the gradients of the mean variables are stronger. The improvement of model performance increases with the increasing complexity of the atmospheric conditions, as for the valley cases, where the enhancements are substantial.
The comparison between the various K–ε closures, differing in the calculation of the countergradient term for the turbulent heat flux, underlines the importance of adopting a prognostic equation for the temperature variance
Finally, this work proves that the new scheme discussed here can improve the reproduction of the atmospheric motion in several conditions, going beyond the definition of a diagnostic turbulent length scale commonly adopted in stateoftheart PBL closures. Future developments will include the validation of the current model in real conditions, through the comparison with observational data, and its coupling with multilayer urban canopy parameterization schemes, in the context of the WRF Model, in order to improve the representation of the complex boundary layer developing over urban areas. Moreover, we aim to extend the herepresented 1D turbulence closure including the horizontal Reynolds stresses, similarly to Juliano et al. (2022), who presented a 3D version of the Mellor and Yamada (1982) turbulence scheme. This effort is expected to improve the performance of NWP models especially over complex terrain, where the horizontal gradients of the turbulent fluxes can be significant, as highlighted by Goger et al. (2018, 2019).
Acknowledgments.
The authors would like to acknowledge highperformance computing support from Cheyenne (https://doi.org/10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation and the Advance Study Program’s Graduate Student (GVP) Fellowship for the financial support.
Data availability statement.
Simulations output (LES and RANS) can be provided by the author upon request, as well as the WRF version including the K–ε turbulence closure. LES of the GABLS case study can be found at https://gabls.metoffice.com/lem_data.html.
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