Abstract
We have identified certain fundamental limitations of a mixinglength parametrization used in a popular turbulent kinetic energybased subgridscale model. Replacing this parametrization with a more physically realistic one significantly improves the overall quality of the largeeddy simulation (LES) of stable boundary layers. For the range of grid sizes considered here (specifically, 1 m–12.5 m), the revision dramatically reduces the gridsize sensitivity of the simulations. Most importantly, the revised scheme allows us to reliably estimate the first and secondorder statistics of a wellknown LES intercomparison case, even with a coarse grid size of O(10 m).
Introduction
The first largeeddy simulation (LES) intercomparison study (Beare et al. 2006), organized under the auspices of the Global Energy and Water Exchanges (GEWEX) Atmospheric Boundary Layer Study (GABLS), has had a lasting impact on research on the stable boundary layer (SBL). In the past decade and a half, the key findings from this study (henceforth referred to as GABLS1–LES) were cited by numerous papers; a few examples are:
“In particular, it is often observed that grid convergence for simulations of the stable boundary layer is lacking, see Beare et al. (2006) and Sullivan et al. (2016). The latter used fine grid spacings down to [0.39] m (pseudospectral code) and still reported a sensitivity of their results to the grid spacing. Until now, a convincing explanation for this behaviour has been lacking, creating a limitation for the application of LES models for simulating the stable boundary layer.” (Maronga et al. 2019)
Similar statements on the gridsize sensitivity can be found in other peerreviewed publications and are often heard in any contemporary workshop or conference session on SBL. Such an overwhelming consensus among the SBL–LES community at large is somewhat disconcerting given the fact that a handful of papers established a while ago that certain dynamic (tuningfree) subgridscale (SGS) models perform rather well (in terms of first and secondorder statistics) with coarse resolutions. As a matter of fact, around the time of publication by Beare et al. (2006), Basu and PortéAgel (2006) demonstrated that the GABLS1–LES case can be simulated reliably by a dynamic SGS parametrization called the locallyaveraged scaledependent dynamic (LASDD) model. They concluded:
“Moreover, the simulated statistics obtained with the LASDD model show relatively little resolution dependence for the range of grid sizes considered here. In essence, it is shown here that the new LASDD model is a robust subgridscale parametrization for reliable, tuningfree simulations of stable boundary layers, even with relatively coarse resolutions.” (Basu and PortéAgel 2006)
Later on, Stoll and PortéAgel (2008) and Lu and PortéAgel (2013) compared different dynamic SGS schemes and also reported negligible sensitivity to grid sizes.
In this study, we revisit the GABLS1–LES case study and probe into the inherent cause of the gridsize sensitivity of a static (nondynamic) SGS parametrization. This parametrization was originally proposed by Deardorff (1980) and appears in many LES codes (e.g., Moeng et al. 2007; Heus et al. 2010; Maronga et al. 2015; Gibbs and Fedorovich 2016). Henceforth, we refer to this as the D80 parametrization. Several past SBL–LES studies have reported gridsize sensitivity with the D80 SGS scheme (e.g., Jiménez and Cuxart 2005; Beare et al. 2006; de Roode et al. 2017; Maronga et al. 2019). After extensive numerical experiments, we have identified the SGS mixinglength (\(\lambda \)) parametrization in this scheme to be at the root of the gridsize sensitivity issue. We have found that a rather simple (yet physically realistic) modification of \(\lambda \) alleviates the gridsize sensitivity substantially. In addition, the first and secondorder statistics from the LES runs utilizing this revised parametrization (named D80R) agrees well with the ones produced by a pseudospectral LES code utilizing a dynamic SGS model. Most importantly, the D80R scheme allows us to reliably simulate the GABLS1–LES case even with a coarse gridsize of O(10 m).
The organization of this paper is as follows: in the following section we describe the D80 parametrization and its fundamental limitations. Physical interpretation and analytical derivations pertaining to the D80R parametrization are included in Sect. 3, and technical details of the simulations are provided in Sect. 4. Results of simulations with D80 and D80R SGS schemes are documented in Sect. 5. A few concluding remarks are made in Sect. 6. In Appendix 1, we have documented a few SGS models which do not use the grid size as a mixinglength scale; instead, similar to the D80R parametrization, they use flowphysicsdependent mixinglengthscale formulations. To further confirm the validity of the D80R approach, results from simulations with an independent LES code are included in Appendix 1.
SGS Parametrization
The SGS parametrization (D80) by Deardorff (1980) utilizes the following mixinglength scale
where \(\varDelta _g = \left( \varDelta x \varDelta y \varDelta z\right) ^{1/3}\) and \(L_b\) is the socalled buoyancy length scale that is typically represented as follows
where \({\overline{e}}\) denotes SGS turbulence kinetic energy (TKE) and N is the Brunt–Väisäla frequency. Deardorff (1980) assumed \(c_n\) to be equal to 0.76. Many LES studies still assume this constant value (e.g., Jiménez and Cuxart 2005; Heus et al. 2010; Maronga et al. 2015) though there are a handful of exceptions. For example, Gibbs and Fedorovich (2016) assumed \(c_n = 0.5\).
The eddy viscosity (\(K_m\)), eddy diffusivity (\(K_h\)), and energy dissipation rate (\({\overline{\varepsilon }}\)) are all assumed to be functions of \(\lambda \) as shown below
The unknown coefficients (\(c_m\), \(c_h\), and \(c_\varepsilon \)) are either prescribed or parametrized as follows
Please note that the values of the coefficients in Eq. 4 somewhat vary between different studies (e.g., Deardorff 1980; Moeng and Wyngaard 1988; Saiki et al. 2000).
For neutral conditions Eq. 1 reduces to \(\lambda = \varDelta _g\). As a direct consequence, \(K_m\), \(K_h\), and \({\overline{\varepsilon }}\) become stability independent, as would be physically expected. Furthermore, SGS Prandtl number (\(Pr_S = K_m/K_h = c_m/c_h\)) is set equal to 0.33 for neutral conditions.
For stably stratified conditions, one expects that \(K_m\), \(K_h\), and \({\overline{\varepsilon }}\) should have a clear dependence on N. Deardorff (1980) accounted for such dependence by introducing \(L_b\) in Eq. 1. Specifically, he stated (we have changed the variable notation for consistency):
“In past work it has been assumed that \(\lambda = \varDelta _g\), which fails to take account of the possibility that in a stably stratified region \(\lambda \) could become much smaller than the grid interval.” (Deardorff 1980)
For the very stable case, in the limit of \(1/N \rightarrow 0\), the D80 SGS model predicts \(\lambda = L_b \rightarrow 0\), \(c_\varepsilon \rightarrow 0.19\), and \(Pr_S \rightarrow 1\). Also, \(K_m\) and \(K_h\) are expected to approach negligibly small values for such conditions.
The behaviour of the D80 SGS model in the weakly and moderately stable boundary layer is rather problematic and is the focus of the present study. For such cases (including the GABLS1–LES case), \(L_b\) is typically on the order of 10 m in the lower and middle parts of the SBL. As mentioned earlier, it has become customary to perform SBL–LES runs with grid sizes of 1–5 m or finer these days. In such simulations, by virtue of Eq. 1, \(\lambda \) becomes equal to \(\varDelta _g\) for lower and middle parts of the SBL. In other words, the ‘min’ operation in Eq. 1 is only utilized in the upper part of the SBL, and in the free atmosphere. Most importantly, akin to the neutral conditions, \(K_m\), \(K_h\), and \({\overline{\varepsilon }}\) become spuriously independent of stability in the lower and middle parts of the SBL.
There is another fundamental problem with Deardorff’s SGS mixinglength parametrization. It is not influenced by the presence of a surface. Deardorff (1980) recognized this problem and proposed a solution of increasing \(c_\varepsilon \) near the surface (see also Moeng 1984). In this context, Gibbs and Fedorovich (2016) stated:
“To this end, it is unclear, however, whether the parameter adjustments incorporated in the original D80 scheme were based on some clear physical reasoning or were intended to merely produce more plausible effects close to the surface.”
To the best of our knowledge, such ad hoc solutions are not implemented in recent LES codes (e.g., Heus et al. 2010; Maronga et al. 2015; de Roode et al. 2017). In other static SGS models (e.g., the Smagorinsky–Lilly model and its variants), empirical wall functions are utilized to explicitly account for the nearsurface shear effects (e.g., Mason and Derbyshire 1990; Brown et al. 1994). Interestingly, it is not a common practice to use wall functions with the D80 SGS scheme. In the case of dynamic SGS models, wall functions are not needed as the estimated SGS coefficient steadily decreases as one approaches any surface, and thus, reduces the SGS mixing length in a selfconsistent manner (e.g., Basu and PortéAgel (2006); Stoll and PortéAgel (2008)).
In the present study, we replace the mixinglength parametrization in D80 (i.e., Eq. 1) with the following formulation
where \(\kappa \) is the von Kármán constant. Both the effects of stability and nearsurface are nicely captured by this equation. In Sect. 3, we will show that the \(\kappa z\) term of this equation can be derived from a wellknown spectral scaling.
The origin of Eq. 5 can be traced back of Blackadar (1962) and Brost and Wyngaard (1978). Blackadar (1962) introduced the following length scale
where, \(\lambda _0\) is an asymptotic length scale. Brost and Wyngaard (1978) and following studies (e.g., Baas et al. 2008) assumed
where, \({\overline{E}}\) is the total TKE.
Please note that despite of apparent similarity, the length scales \(L_b\) and \(L_b^*\) are quite different. Scale \(L_b\) is proportional to SGS TKE (\({\overline{e}}\)), and thus implicitly depends on the filter size \(\varDelta _f\) (see Sect. 3). In contrast, \(L_b^*\) is used in Reynoldsaveraged Navier–Stokes (RANS) models, and by definition, it does not depend on \(\varDelta _f\).
Hereafter, we refer to Deardorff’s SGS parametrization in conjunction with Eq. 5 as the D80R parametrization. In this parametrization, with increasing resolution, both \({\overline{e}}\) and \(L_b\) decrease. As a result, \(\lambda \), \(K_m\), and \(K_h\) also decrease. Additional effect of gridsize is also felt via \(c_h\), and \(c_\varepsilon \) coefficients. In terms of SGS Prandtl number (\(Pr_S\)), both the D80 and D80R parametrizations suffer from unphysical prescription in different ways. This issue is discussed in detail in Sect. 5.
Physical Interpretation and Analytical Derivation
In this section, we make direct associations between the energy spectra of turbulence and several key elements of the D80R parametrization. In addition, we introduce a generalized form of Eq. 5 which has the potential to further extend the regime of application of the proposed D80R parametrization.
Dependence of \({\overline{e}}\) on Filter Size (\(\varDelta _f\))
A simple model of longitudinal velocity spectrum, spanning all the scales of turbulence, can be written as (see page 232 of Pope 2000)
where k is wavenumber, \(E_u\) is the energy spectrum for longitudinal velocity component, \(L_I\) and \(\eta \) denote integral length scale and Kolmogorov’s scale, respectively. The nondimensional functions \(\varPhi _{L_I}\) and \(\varPhi _{\eta }\) represent the buoyancy range and dissipation range, respectively. In the inertial range, both these functions are close to unity. For small values of \(k L_I\) and large values of \(k \eta \), they deviate from unity.
If the filter size \(\varDelta _f\) is within the inertialrange (as inherently assumed in LES), the subgridscale variance of longitudinal velocity component \(\sigma _{us}^2\) can be estimated from Eq. 8 as
where, \(k_\eta \) is the dissipation wavenumber. The wavenumber associated with the filter is \(k_\varDelta = \frac{\pi }{\varDelta _f}\). Within the range of \(k_\varDelta \) to \(k_\eta \), \(\varPhi _{L_I}\) is unity. Since the contribution of dissipation range is typically small towards \(\sigma _{us}^2\), one can assume \(\varPhi _{\eta } \approx 1\) in Eq. 9.
Owing to isotropy in the inertial range, Eq. 9 can be integrated as follows
Since \(k_\eta \gg k_\varDelta \), we can simplify this equation as
Eq. 11 can be rewritten as
This equation is analogous to Eq. 3c, if one replaces \(\lambda \) with \(\varDelta _f\).
Estimation of \({\overline{e}}\) in Surface Layer
The derivations in Sect. 3.1 assume \(\varDelta _f\) falls within the inertial range. However, in the surface layer, the inertial range is rather limited. For an extensive range of scales, the longitudinal velocity spectrum follows a \(k^{1}\) power law. Thus, in most LES studies, it is likely that \(\varDelta _f\) falls within the \(k^{1}\) range in the surface layer and not within the inertial range.
Following Tchen (1953), numerous studies have reported the \(k^{1}\) scaling in the literature; please refer to a comprehensive list in Table 1 of Katul and Chu (1998). By combining the \(k^{1}\) scaling with the inertialrange scaling (i.e., \(k^{5/3}\)), the energy spectrum for the longitudinal velocity spectrum can be written as
where the friction velocity is denoted by \(u_*\). The crossover wavenumber \(k_o\) equals to \(\frac{1}{\gamma z}\). For unstable conditions, \(\gamma \) was found to be equal to unity by Kader and Yaglom (1991) and others. For stable conditions, the value of \(\gamma \) decreases from unity with increasing stability (Banerjee et al. 2016).
By integrating Eq. 13, we can estimate the SGS variance of longitudinal velocity component as follows
where, \(c_1\) and \(c_2\) are unknown constants. Due to strong anisotropy in the surface layer, we cannot estimate SGS TKE (\({\overline{e}}\)) from \(\sigma _{us}^2\). However, it is expected that \({\overline{e}}\) will be proportional to \(\sigma _{us}^2\).
The first integration term of Eq. 14 can be simplified as
Whereas the second term of Eq. 14 amounts to
Please note that, according to Eq. 16, \(\sigma _{us2}^2\) is not dependent on \(\varDelta _f\), rather it is dependent on z. Due to logarithmic operation in Eq. 15, \(\sigma _{us1}^2\) weakly depends on \(\varDelta _f\). Thus, \({\overline{e}}\) is expected to be weakly dependent on \(\varDelta _f\) in the surface layer. More importantly, Eq. 12 cannot be used to estimate \({\overline{\varepsilon }}\) in the surface layer.
In general, both the terms \(\sigma _{us1}^2\) and \(\sigma _{us2}^2\) contribute to \({\overline{e}}\). It is not possible to neglect either of them for further simplification. However, if we assume that both these variances are proportional to \({\overline{e}}\), we get qualitatively similar results regarding the mixing length.
Based on Eq. 16, we can write
and
where, \(c_\gamma \) is an unknown constant. By comparing Eq. 3c with Eq. 17b, we can assert that the mixing length scale \(\lambda \) in D80 and D80R should be proportional to z in the surface layer instead of \(\varDelta _f\). In the following section, we arrive at the same conclusion via a different route.
ShearBased Mixing Length Scale
Let us assume that S represents the magnitude of velocity shear. Thus, gradient Richardson number (\(Ri_g\)) equals to \(N^2/S^2\). For sheardominated flows, Hunt et al. (1988, (1989) proposed a length scale
where \({\overline{E}}\) is the total TKE. It is a common knowledge that the effects of shear are more prevalent than the buoyancy effects near the surface. For such a situation, \(L_H^*\) is a more relevant length scale than \(L_b^*\) (defined earlier in Eq. 7).
For SGS modelling, if \({\overline{E}}\) is replaced with \({\overline{e}}\), an analogous shearbased length scale can be defined
In contrast to \(L_H^*\), the length scale \(L_H\) implicitly depends on \(\varDelta _f\) due to its explicit relationship to \({\overline{e}}\).
Earlier we showed that Eq. 15 holds near the surface. Now, if we assume \({\overline{e}}\) is proportional to \(\sigma _{us1}^2\), we can combine Eq. 15 with Eq. 19 and obtain
where \(\phi _m\) is the nondimensional velocity gradient.
For finitedifferencebased LES codes, \(\varDelta _f\) is typically 4–6 times larger than \(\varDelta _g\). For isotropic grids, the height of vertical levels (z) equal to \(m \varDelta _g\); where, m are halfintegers. For vertically stretched grids, z could be a small fraction of \(\varDelta _g\) for the first few levels. For stably stratified conditions, \(\phi _m \ge 1\). As mentioned earlier, in such situation \(\gamma \le 1\). Based on these estimates, it is reasonable to state that \(\lambda \) is proportional to z in the surface layer; the exact value of the proportionality constant is unknown. In this study, based on other usage described in the literature (specifically in RANS modelling), we assumed the proportionality constant to be equal to \(\kappa \).
A Generalized MixingLength Parametrization
The proposed D80R mixinglength parametrization (i.e., Eq. 5) is valid for stable conditions. However, for nearneutral conditions, as N approaches zero, the value of \(\lambda \) could become very large. To account for such stability regimes, one can adopt a more generic parametrization for the mixing length. Essentially, one can combine \(L_H\) and \(L_b\) in the following manner
where the unknown coefficients \(c_H\) and \(c_b\) should be prescribed. For nearneutral condition, the term involving \(L_H\) will dominate and will lead to realistic \(\lambda \) values. As discussed in the previous section, this term will also perform well in the surface layer.
Unfortunately, we do not know how to estimate optimal values of \(c_H\) and \(c_b\) in a meaningful way. Running numerous largeeddy simulations for various case studies with different combinations of \(c_H\) and \(c_b\) is not a computationally viable option. Hopefully, an efficient strategy will emerge in the near future. For the timebeing, we utilize an approximation of Eq. 21, i.e., Eq. 5, as a working substitute for SBL simulations.
Description of the Simulations
In this work, we simulate the GABLS1–LES case study using the Dutch Atmospheric LargeEddy Simulation (DALES; Heus et al. 2010) and the PALM model system (Maronga et al. 2015, 2020). Since the configurations of the GABLS1–LES case study are well described in the literature, we mention them in a succinct manner. The boundary layer is driven by an imposed uniform geostrophic wind speed of 8 m s\(^{1}\) and a surface cooling of 0.25 K h\(^{1}\). It attains a quasisteady state in about 8–9 h with a depth of approximately 200 m. The initial mean potential temperature is 265 K up to 100 m with an overlying inversion of 0.01 K m\(^{1}\). The Coriolis parameter is set to \(1.39 \times 10^{4}\) s\(^{1}\), corresponding to latitude 73\(^{\circ }\) N. Both the aerodynamic roughness length \(z_0\) and the scalar roughness length \(z_{0h}\) are assumed to be equal to 0.1 m.
For all runs, the computational domain is fixed at 400 m \(\times \) 400 m \(\times \) 400 m. A wide range of isotropic \(\varDelta _g\) values are used to investigate the aforementioned gridsize sensitivity issue. For the DALES code, in order to avoid any temporal discretization error, the time step \(\varDelta t\) is kept at a constant value of 0.1 s for all the simulations. In contrast, an adaptive timestepping approach is used by the PALM model system. In addition to the results from the finitedifferencebased DALES and PALM codes, we also report results from a pseudospectral code (called MATLES) utilizing the LASDD SGS model along with a grid size of 3.5 m and a fixed timestep of 0.075 s.
In terms of the numerical schemes in the DALES and PALM codes, a thirdorder Runge–Kutta scheme is used for time integration in conjunction with a fifthorder advection scheme in the horizontal direction (Wicker and Skamarock 2002). In the vertical direction, a secondorder and a fifthorder scheme (which reduces to a secondorder scheme near the surface) are used by the DALES and PALM codes, respectively. The MATLES code uses a secondorder Adams–Bashforth scheme for time advancement.
In all the simulations, the lower boundary condition is based on the Monin–Obukhov similarity theory (MOST). As discussed by Basu and Lacser (2017), in order to apply MOST, the first model level in an LES model should not be at a height lower than \(\sim 50 z_0\). In this study, for simplicity, this requirement has been violated for all the runs involving highresolution. At this point, the impact of this violation on the simulated statistics is not noticeable and a thorough investigation on surfacelayer physics will be conducted in a followup study.
Before discussing the results, we point out that some of the prescribed SGS constants differ between the DALES and the PALM codes. For example, \(c_m\) is assumed to be equal to 0.12 in DALES; whereas, PALM sets it at 0.1. The other specifications related to \(c_h\) and \(\lambda \) depend on local \(N^2\) values and are listed in Table 1. In addition, the coefficients in Eq. 4c are also slightly different in DALES and PALM. In spite of these differences, the results from DALES and PALM codes follow the same trends as depicted in the following section and in Appendix 1.
Results
The left and right panels of all the following figures represent the statistics from the D80 and D80Rbased runs, respectively. As prescribed in the GABLS1–LES study, all the statistics are averaged over the last one hour (i.e., 8–9 h) of the simulations.
Prior to discussing the first and secondorder statistics, it is important to highlight the differences between the original and the revised runs in terms of the SGS mixinglength (\(\lambda \)) profiles. From Fig. 1 (top panel), it is evident that in the D80based runs (left panel), \(\lambda \) equals to \(\varDelta _g\) for the lower and middle parts of the SBL. In contrast, the D80Rbased runs show clear dependence on the distance from the surface. In both types of runs, \(\lambda \) values monotonically decrease to zero in the upper part of the SBL and in the free atmosphere due to increasing stratification (as quantified by N). The \(\lambda \) profiles from the D80Rbased simulations also show clear dependence on grid size in the core region of the SBL. This is due to the fact that SGS TKE decreases with increasing resolution, and in turn, decreases \(\lambda \).
In the D80based runs, due to the underlying condition of \(\lambda = \varDelta _g\), \(Pr_S\) becomes exactly equal to 0.33 for the lower part of the SBL (bottomleft panel of Fig. 1). Whereas, in the case of D80R, \(\lambda \) is typically smaller than \(\varDelta _g\) near the surface. Thus, \(Pr_S\) is much larger than 0.33. However, in the middle part of the SBL, depending on stratification, \(\lambda \) may become larger than \(\varDelta _g\). For such cases, in D80Rbased runs, \(Pr_S\) can be even smaller than 0.33 (bottomright panel of Fig. 1). In the upper part of the SBL, due to stronger stratification, \(\lambda \) becomes much smaller than \(\varDelta _g\), and as a consequence, \(Pr_S\) monotonically approaches to unity for both the D80 and D80R cases. In the case of the LASDD SGS model, the dynamically estimated \(Pr_S\) values remain more or less constant (around 0.5–0.6) inside the SBL and becomes greater than 1 in the inversion layer.
Gibbs and Fedorovich (2016) recognized the problem with the \(Pr_S\) in D80 parametrization. They proposed to use \(Pr_S = 1\) when \(N^2\) is locally positive. For locally negative \(N^2\) values they proposed an empirical formulation for \(Pr_S\). In the present study, thus, the PALM model is employed with \(Pr_S = 1\) for \(N^2 > 0\) in all the D80Rbased runs. These results are shown in Appendix 2. Please note that for \(N^2 \le 0\), both the DALES and the PALM models always use Eq. 4b (i.e., they effectively assume \(Pr_S\) = 0.33); we have not incorporated the empirical formulation by Gibbs and Fedorovich (2016).
Simulated time series of surface friction velocity and sensible heat flux are documented in Fig. 2. In the D80based runs, no temporal fluctuations of surface fluxes are visible for \(\varDelta _g \ge \) 6.25 m. When \(\varDelta _g\) is finer than 6.25 m, temporal fluctuations do appear approximately 1–2 hours into the simulation. Increasing gridresolution helps in systematically reducing this turbulence kickoff time. In the D80Rbased runs, similar artifacts are not visible in the dynamical evolution of surface fluxes. In the D80based runs for \(\varDelta _g \ge \) 6.25 m, \(\lambda \) (= \(\varDelta _g\)) values are excessively large near the surface and simply do not allow for the sustenance of turbulence. The inclusion of the surface dependence term in Eq. 5 reduces the mixing length in a meaningful way and promotes the production and transport of turbulence.
In Fig. 2, the simulated time series from the MATLES code are also overlaid. In the D80Rbased runs, the agreement between the DALES and MATLESbased results is remarkable. All the time series of surface friction velocity reach quasiequilibrium stage after approximately 5 h of simulation. The surface sensible flux time series reach quasiequilibrium stage at a later time (\(\sim \) 6 h).
The vertical profiles of mean wind speeds are included in the top panel of Fig. 3. The presence of the lowlevel jet (LLJ) is clearly visible in both the plots. However, the height of the LLJ peak shows strong sensitivity with respect to gridsize in the D80based runs when \(\varDelta _g \ge \) 6.25 m. This unphysical behaviour is solely due to the inherent limitations of the D80 SGS parametrization. Once \(\varDelta _g\) becomes smaller than 6.25 m, turbulence is sustained near the surface, and as a result of adequate diffusion, the simulated LLJ peak heights are elevated. With further increase in spatial resolution, the mean windspeed profiles reach convergence; albeit, the height of the DALESbased simulated LLJ peak remains lower and weaker than the one simulated by the MATLES code. The positive impacts of the revised mixinglength parametrization in D80R can be seen in the topright panel of Fig. 3. Almost all the DALESbased runs (with the exception of the one with \(\varDelta _g = \) 12.5 m) converge on a single line and also agree strongly with the MATLESbased results.
Utilizing the results from the GABLS1 intercomparison study, Svensson and Holtslag (2009) investigated the wind turning in SBL in great detail. With certain assumptions, they analytically derived the relationship between the vertically integrated crossisobaric flow and surface friction. Earlier, we have shown that the runs using the D80R parametrization lead to \(u_*\) series which are insensitive to grid sizes. Thus, it is not surprising that those simulations also lead to wind direction profiles which are in good agreement with one another. From the middle panel of Fig. 3, it is also clear that the original D80 parametrization performs quite poorly in capturing the turning of wind with height for coarse grids.
The profiles of mean potential temperature are shown in the bottom panel of Fig. 3. All the profiles do show similar convex curvature as reported by earlier studies on GABLS1–LES. However, as with the mean windspeed profiles, the D80based runs exhibit strong dependence on grid size. Once again, incorporation of the revised SGS mixing length parametrization leads to better convergence. However, in this case, lower values of \(Pr_S\) in the middle part of the SBL (see bottomright panel of Fig. 1) cause more heat diffusion; not surprisingly, the potential temperature profiles from the D80Rbased runs are more convex than the one simulated by the MATLES code. In contrast, by using a higher value of \(Pr_S\), the PALM model generates potential temperature profiles which are indistinguishable from the MATLESbased ones (refer to Fig. 11 in Appendix 2).
In the D80based runs with finer resolutions, the simulated mean potential temperature profiles also start to converge. However, for these cases, \(Pr_S\) values are equal to 0.33 in the lower part of the SBL. The (negative) impact of such high eddy diffusivity values is hard to notice in the mean potential temperature profiles; however, the vertical profiles of variance of potential temperature, \(\sigma _\theta ^2\), do show the effect clearly (discussed later).
The x and ycomponents of momentumflux profiles are shown in Figs. 4 and 5, respectively. Several observations can be made from these plots. First of all, the resolved momentum fluxes are virtually nonexistent for the D80based runs with \(\varDelta _g \ge 6.25\) m. This result is not surprising given the other statistics shown in the previous plots. The total fluxes are strongly gridsize dependent for the D80based runs; however, they are not for the D80Rbased ones. Also, the fluxes are well resolved in the revised case; near the surface, the resolved fluxes increase with increasing resolution as would be expected. The revised results are more or less in line with the fluxes generated by the MATLES code.
As depicted in Fig. 6, the overall behaviour of the total and resolved sensible heat flux profiles are qualitatively similar to those of the momentumflux profiles. In the case of the D80Rbased runs, the grid convergence is slightly less than satisfactory for the total sensible heatflux profiles. Especially, the simulation with \(\varDelta _g = 12.5\) m consistently overestimates the magnitude of heat flux across the SBL.
The resolved and SGS TKE plots are shown in Fig. 7. The MATLES code does not solve for the SGS TKE equation; thus, only the resolved TKE values are overlaid for comparison. As with the momentumflux and sensible heat flux profiles, the resolved TKE is nonexistent in the bottomhalf of the SBL for the D80based simulations using \(\varDelta _g \ge \) 6.25 m. In the lower part of the SBL, the resolved TKE values are larger in the D80Rbased runs in comparison to the D80based ones. In that region, with increasing resolution, resolved TKE values increase as would be physically expected. Most importantly, the SGS TKE values are much smaller in magnitude than their resolved counter parts (especially, in the lower part of the SBL). In other words, the flow is highly resolved for all the simulations involving the D80R parametrization.
Lastly, the resolved variances of vertical velocity and potential temperature are plotted in Fig. 8. The trends of the resolved \(\sigma _w^2\) profiles are similar to the resolved TKE plots. The resolved \(\sigma _\theta ^2\) profiles from the D80based runs are quite interesting. As discussed earlier, the coarseresolution runs (i.e., \(\varDelta _g \ge \) 6.25 m) show the quasilaminarization problem up to 75 m or so. Interestingly, for higher resolution runs with the original mixinglength parametrization, the resolved \(\sigma _\theta ^2\) values decrease significantly near the surface. The opposite trend is seen in the D80Rbased simulations. We believe this decrease in the D80based runs is due to the low value of \(Pr_S\) (= 0.33) in the bottom part of the SBL. Similar decrease in \(Pr_S\) in the core of the SBL also causes resolved \(\sigma _\theta ^2\) to significantly decrease in the D80Rbased cases.
Even though most vertical profiles from the DALES code look physically realistic and agree quite well with the corresponding results from the MATLES code, some discrepancies are noticeable in the case of resolved variances and fluxes. It is possible that the LASDD SGS model is slightly overdissipative near the surface; the alternative scenario is that the proposed D80R SGS model is slightly underdissipative near the surface. Typically, in pseudospectral codes, spectral analysis can shed light into such undesirable behaviours (see Anderson et al. 2007 for some examples). Spurious piling up of energy near the highwavenumber part of the spectrum is a telltale sign of underdissipation. Since DALES and PALM are finitedifference codes, detection of such a subtle feature in the energy spectrum is a challenging task due to strong numerical dissipation. As such, we will be implementing the D80R SGS model in the MATLES code for indepth spectral analysis.
Discussion
In addition to Deardorff’s SGS model, the Smagorinsky–Lilly SGS model (Smagorinsky 1963; Lilly 1966a, b) is also quite popular in the boundarylayer community. In this SGS model, the effective mixing length is \(C_S \varDelta _f\); where, \(C_S\) is the socalled Smagorinsky coefficient. This coefficient is adjusted empirically (e.g., Mason and Derbyshire 1990; Brown et al. 1994; Kleissl et al. 2004) or dynamically (e.g., PortéAgel et al. 2000; BouZeid et al. 2005; Basu and PortéAgel 2006) to account for shear, stratification, and grid resolution.
In contrast, the effective mixing length is \(c_m \lambda \) in D80 SGS model. Most of the LES codes assume a constant value for \(c_m\) and expect \(\lambda \) to account for shear, stratification, and grid resolution. As elaborated in Sect. 2, the original mixinglength formulation does not account for shear or stratification properly. The parametrization by Brost and Wyngaard (1978) provides a major improvement as both the shear and stratification effects are now explicitly included. The effects of grid size is somewhat indirect. The performance of the D80R SGS model may be improved if \(c_m\) is assumed to be \(\varDelta _g\)dependent and estimated via a dynamic approach (e.g., Ghosal et al. 1995; Krajnović and Davidson 2002). Our future work will be in that direction.
We will also need a better understanding of the SGS Prandtl number (\(Pr_S\)) and its relationship to the turbulent Prandtl number (\(Pr_T\)). Several field and laboratory experiments have reported that \(Pr_T\) should be on the order of one for stably stratified conditions (see Sukoriansky et al. 2006; Li 2019, and the references therein). However, within the SBL (excluding surface and inversion layers), the dynamic SGS models typically estimate \(Pr_S\) and \(Pr_T\) values to be around 0.5–0.6 and 0.7, respectively (Basu and PortéAgel 2006; Stoll and PortéAgel 2008). We agree with Li (2019) that \(Pr_S\) is less important than \(Pr_T\) as the largescale fluxes are resolved in LES. Nonetheless, we do believe the current \(Pr_S\) parametrizations in the D80 and D80R SGS models are not acceptable and definitely need amendments. The formulation proposed by Gibbs and Fedorovich (2016) is a good starting point and should be rigorously tested in future studies.
Concluding Remarks
In this study, we have demonstrated that Deardorff’s mixinglength parametrization is not suitable for SBL simulations. Instead an older scheme, proposed by Brost and Wyngaard for RANS, gives promising results. Even though we have made some progress in the arena of LES of SBL, many open questions still remain:

Would this new mixinglength parametrization work for very stable boundary layers? Would it allow us to simulate turbulent bursting events?

Is the proposed parametrization underdissipative near the surface? How can we (dis)prove this behaviour?

Is a buoyancybased length scale really appropriate for weakly or moderately stable boundary layer? Or, should we opt for a shearbased length scale (Hunt et al. 1988, 1989; Basu et al. 2020)?

How sensitive are the simulated results with respect to the choice of SGS coefficients (i.e., \(c_n\), \(c_m\), \(c_\varepsilon \))? Should they be dynamically determined instead of being prescribed?

How should we parametrize the SGS Prandtl number? Should it be a function of pointwise gradient Richardson number?
A few years ago, Basu and Lacser (2017) cautioned against violating MOST in LES runs with very high resolutions. To overcome this issue, Maronga et al. (2019) proposed certain innovative strategies; however, the results were somewhat inconclusive—(possibly) due to the usage of the D80 mixinglength parametrization in all their simulations. In light of the findings from the present work, we will revisit the MOST issue in very high resolution LES in conjunction with the D80R SGS model. In addition, we will investigate the interaction of the D80R parametrization with a coupled landsurface model.
We further recommend the SBL–LES community to revisit LES intercomparison studies organized under GABLS with revised or newly proposed SGS models. We speculate that some of the conclusions from the previous studies will no longer be valid.
Data and Code Availability
The DALES code is available from: https://github.com/dalesteam/dales. The PALM model system is available from: https://palm.muk.unihannover.de/trac. The MATLES code is available from S. Basu upon request. Upon acceptance of the manuscript, all the analysis codes and processed data will be made publicly available at http://doi.org/10.5281/zenodo.3972345.
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Acknowledgements
The authors are grateful to Patrick Hawbecker, Bert Holtslag, Branko Kosović, Arnold Moene, Siegfried Raasch, and Chiel van Heerwaarden for useful discussions.
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Appendices
Appendix 1: SGS Models with Implicit \(\varDelta _f\)Dependence
Traditional eddyviscosity models, commonly used by the boundarylayer meteorology community, explicitly involve the filtersize (\(\varDelta _f\)) as the mixinglength scale. For example, in the case of Smagorinsky–Lilly SGS model, one uses
where \({\tilde{S}}\) is the magnitude of the resolved strain rate tensor and \(C_S\) is known as the Smagorinsky coefficient.
In contrast to these popular SGS models, the proposed D80R SGS model depends on \(\varDelta _f\) in an implicit manner. Similar implicit \(\varDelta _f\)dependence can be found in the engineering LES literature. Recently, Piomelli and Geurts (2011) proposed an eddyviscosity model
where, \({\overline{e}}_r\) and \(\omega _r\) denote resolved TKE and resolved magnitude of vorticity, respectively. In a more recent work, Piomelli et al. (2015) formulated the integral length scale approximation (ILSA) model
where \({\overline{\varepsilon }}_T\) represents the total energy dissipation rate. A dynamic version of the ILSA model was developed by Rouhi et al. (2016) and utilized for simulations in complex geometries by Lehmkuhl et al. (2019). The advantages of these flowphysicsdependent length scales (i.e., Eq. 23b and Eq. 24b) over \(\varDelta _f\) as a mixinglength scale have been extensively discussed by Piomelli (2014) and Geurts et al. (2019) and will not be repeated here for brevity.
In addition to the aforementioned eddyviscosity type SGS models, certain noneddyviscosity type SGS models also do not include explicit dependency on \(\varDelta _f\). A case in point is the similarity model and its variants (Bardina et al. 1980; Liu et al. 1994).
Appendix 2: Results from the PALM Model System
The simulated results from the PALM model are documented in this appendix (see Figs. 9, 10, 11, 12, 13, 14, 15 and 16). The trends are very similar to those reported in Sect. 4. Thus, we do not discuss most of these figures for brevity. However, we would like to point out a noticeable feature in the D80based runs. Even though most of the simulated profiles converge for 2 m \(\le \varDelta _g \le \) 5 m, the results from \(\varDelta _g = 1\) m run stands out. We believe that \(\lambda \) is quite low for this particular case; even though it is small enough to sustain turbulence near the surface, it is not large enough to promote diffusion. The D80R run with \(\varDelta _g = 1\) m does not portray such an unusual behaviour.
The (positive) impacts of higher SGS Prandtl number (\(Pr_S\)) utilized in the D80Rbased runs can be seen in some of the figures. First of all, Fig. 11 (bottomright panel) shows that the vertical profiles of potential temperature are less convex than the corresponding profiles from the DALES code (i.e., bottomright panel of Fig. 3). This is due to less heat diffusion. For the very same reason, the PALM modelgenerated resolved \(\sigma _\theta ^2\) values are much larger than their DALES counterpart (see Fig. 16).
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Dai, Y., Basu, S., Maronga, B. et al. Addressing the GridSize Sensitivity Issue in LargeEddy Simulations of Stable Boundary Layers. BoundaryLayer Meteorol 178, 63–89 (2021). https://doi.org/10.1007/s10546020005581
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Keywords
 Buoyancy length scale
 Prandtl number
 Stable boundary layer
 Subgridscale model