1. Introduction
The object of this research is to study the exchange of heat and momentum between atmospheric turbulence and surfaces with large roughness elements. We consider urbantype surfaces because, for them, a significant thermal stratification effect on turbulence in a layer filled with streamlined objects is expected. In addition, such surfaces are relatively easy to specify explicitly in the LES model to be used for research. First, we will be interested in the heat transfer and aerodynamic properties of such surfaces as a whole and the possible influence of buoyancy forces on these properties. We will restrict ourselves to the cases of neutral and stable stratification. Turbulent flows in a state of statistical equilibrium are considered.
In cases where the characteristic height of the roughness elements 〈h〉 cannot be considered negligibly small compared to z, Eq. (2) remain an acceptable approximation, with the only difference that instead of the height z we should use the distance to some imaginary surface z′ = z − D, where D is the displacement height (a dimensional parameter that characterizes the effect of the geometric characteristics of streamlined objects on the turbulent length scales of external flow and usually D < 〈h〉).
At very high Reynolds numbers, Re ≫ 1 (if the thickness of the viscous sublayer is negligibly small compared to the characteristic sizes of the roughness elements), the total surface frictional stress is mainly due to the form drag of the objects. In this case, with nearly neutral stratification 〈h〉 ≪ L, the dynamic roughness length z_{0}_{u} is determined only by the geometric properties of the surface and does not depend on Re. The value of z_{0}_{u} can be obtained from measurement data or numerical simulation results. Then, different surfaces are classified by their shape, and the value of z_{0}_{u} may be estimated using a finite set of surface geometry parameters, e.g., the average height of the roughness elements, the variance of their heights, the average distances between the objects. For the urbanlike canopies considered below, z_{0}_{u}/〈h〉 ∼ 10^{−2}.
The greater difficulty lies in the estimation of the thermal roughness length z_{0}_{t}. There is no form drag for the heat flux, which implies z_{0}_{t} ≪ z_{0}_{u}, so that the heat exchange occurs, ultimately, due to the molecular heat transfer in a thin nearsurface layer on the complex surface of streamlined objects. Accordingly, z_{0}_{t} depends not only on the geometry of the roughness elements, but also on the friction velocity
In Zilitinkevich et al. (2008a) the dependencies of the dynamic roughness length z_{0}_{u} and displacement height D_{u} on the parameter 〈h〉/L were proposed. According to Zilitinkevich et al. (2008a), z_{0}_{u} decreases, while D_{u} increases in the stable atmospheric boundary layer (ABL); the opposite effect is observed under unstable stratification. This parameterization shows good qualitative and quantitative agreement with measurements over forest vegetation (Joffre et al. 2001; Gryning et al. 2001). However, according to the LES results (Glazunov 2014c), no significant changes in z_{0}_{u} and D_{u} values at h/L = 1 with respect to their values under neutral stratification were found. Probably, this is related to the fact that the downward turbulent momentum transfer within the canopy, on which the reasoning in Zilitinkevich et al. (2008a) was based, plays an equivalent or even a minor role in an urban environment with respect to the dispersive momentum flux formed by timestationary eddies [see the definition of dispersive flux in Poggi et al. (2004)], and also because of the significant role of the aerodynamic form drag from the “buildings,” which contribution was omitted in this work.
The use of numerical models (LES and DNS) to identify the aerodynamic characteristics of urbanlike surfaces is not new. The first such simulations were performed about 15 years ago (e.g., Xie and Castro 2006; Coceal et al. 2006) and showed good agreement with laboratory data (see Cheng and Castro 2002). Modern supercomputing facilities allow not only a much wider range of surface shapes to be considered, but also the turbulent flow characteristics within the 0 < z < h layer to be accurately studied (see, e.g., Cheng and PortéAgel 2015). This, in particular, allows LES data to be used to develop and to refine multilayer onedimensional RANS models of urban ABL (Nazarian et al. 2020; Cheng and PortéAgel 2021; Glazunov et al. 2021).
Numerical simulations of stably stratified urban ABL are quite rare (e.g., Glazunov 2014b,c; Cheng and Liu 2011; Xie et al. 2013; Boppana et al. 2014; Tomas et al. 2016; Shen et al. 2017; Grylls et al. 2020). In most of the abovementioned works, such simulations were carried out with relatively weak stability. However, the highest concentrations of pollutants near the surface are achieved under low wind and for strong stable stratification [see, e.g., the review by Li et al. (2021)]. Therefore, one can consider a stably stratified urban ABL as one of the most important cases for research and parameterization. Additional problems related with using LES models to study steadystate urban ABLs are the complexity of setting up a numerical experiment to simulate quasiequilibrium regimes and the need for longer simulations compared to integration time in neutral conditions [see Grylls et al. (2020, p. 328) for these and some other challenges in such numerical experiments]. Here we present an algorithm that overcomes these difficulties and can facilitate studies of the stably stratified urban boundary layer.
Note that extensive wind tunnel measurements of stably stratified turbulence in urbanlike environments exist, e.g., measurements of flow over and inside an array of cubes (Uehara et al. 2000) or measurements of flow over an isolated street canyon (Marucci and Carpentieri 2019). In these experiments, turbulent flow is generated over an extended area with small roughness elements before approaching urbantype objects. Correct quantitative comparisons of numerical simulations with such measurements require a special setup with large computational domains, replicating the configuration of wind tunnels. In addition, straight reproduction of the experimental setup in a numerical setting requires very fine grids to resolve the molecular heat transfer at the surfaces with a given temperature. Although direct comparison of the presented simulations with such measurements is not possible, some qualitative features of stably stratified urban turbulent flows appear to be common to both numerical experiments and laboratory ones. It will be noted in the following sections.
In Glazunov (2014b,c) LES of stable urbanlike ABL at values L ≈ h were carried out. The primary focus was on turbulent momentum transfer at heights z > h. The flow inside the urban canopy was not analyzed because of the coarse spatial resolution, which does not accurately reproduce the turbulent flow around the objects. In addition, the design of the numerical experiments (Glazunov 2014c,b) did not allow one to calculate the values of z_{0}_{t} because the surface heat flux Q_{s} rather than the surface temperature T_{s} was prescribed.
Here we present a series of similar experiments with increased spatial resolution of the LES model and with an extended range of stability parameter 〈h〉/L values. In addition, we modified the setup of numerical experiments in such a way that the equilibrium state of the turbulent flow for a given value of 〈h〉/L is achieved by changing the surface temperature T_{s} with time. In this case, the heat flux Q_{s}, which depends on the local instantaneous values of the nearsurface air velocity u_{w} and temperature T_{w}, is calculated. This allows one to pose the problem of identifying the relation between the thermal roughness length z_{0}_{t} of the complex surface and the value of the stability parameter 〈h〉/L, while revealing the underlying physical origins of this dependence.
2. Setup and parameters of numerical experiments
a. LES model
In detail this model is described in Glazunov et al. (2016). This model allows the use of relatively coarse spatial grids and prevents early laminarization by using energyconservative high (fourth)order spatial approximations for momentum and scalars (Morinishi et al. 1998) and localized mixed dynamic (Germano et al. 1991) subgrid/subfilter closure.
Earlier this LES model has been tested for neutrally stratified turbulent flows over urbantype surfaces and has shown the ability to correctly reproduce some integral characteristics of such flows, e.g., the displacement height D_{u} and roughness length z_{0}_{u} of the staggered cubes array (Glazunov 2014a) and the mean characteristics of the scalar turbulent transport inside the streetcanyons canopy (Glazunov 2018). For stably stratified conditions the LES model has only been tested for flat surfaces [see grid sensitivity tests (Glazunov 2014c) and comparison with other LES models (Glazunov et al. 2016) under the GABLS1 scenario (Beare et al. 2006), and as well as comparison with DNS results for a turbulent stably stratified plane Couette flow (Glazunov et al. 2019)].
b. Simplifications and assumptions accepted
Note that the spatial and temporal resolution of modern numerical models is still insufficient to approximate the entire range of scales of realscale urban turbulence. In particular, the thin nearwall laminar sublayers through which the molecular heat transfer occurs are not reproduced explicitly in the simulations which will be discussed. Indeed, for the characteristic values of the wind speed inside the canopy U_{w} ∼ O(1) m s^{−1} and the corresponding characteristic values of the nearwall friction velocity
In addition, when calculating the instantaneous heat fluxes and frictional stresses on various surfaces (on the “walls” and “roofs” or on the “ground”) we cannot rightfully use the expressions based on MOST [e.g., (4)], because the universal form of the coefficients C_{U} and C_{T} is valid only for flows with horizontal homogeneity. Therefore, we fix the values
Taking into account the mentioned specifics of the numerical experiment setup, we draw the reader’s attention to the fact that the calculated values of the thermal roughness length z_{0}_{t} and inverse Stanton number
c. Domain geometries and boundary conditions
Here we present the results of 13 LES runs performed on an equidistant grid. The runs are divided into three groups EXP1, EXP2, and EXP3 (see Figs. 1a–c, respectively) according to the geometry and location of objects on the lower boundary of the computational domain (the cubes and rectangular parallelepipeds imitating buildings). According to terminology introduced by Cheng and Castro (2002) all of the simplified urbanlike surfaces considered may be classified as “aligned” arrays (see difference between “staggered” and “aligned” arrays in the mentioned paper). In Figs. 1a–1c, h and h/2 are the heights of the objects (here, h = 32Δ, where Δ is grid step). The horizontal sizes of the objects were equal to h in EXP1 and EXP2 and h/2 in EXP3. Accordingly, the plan area density λ_{p}, which is the fraction of the ground surface covered by the objects, was equal to 0.0625 in EXP1 and EXP2 and 0.015 625 in EXP3. The surface geometry in group EXP1 corresponds to the case C20S6.25% of wind tunnel measurements in Cheng et al. (2007).
The size of the entire computational domain was L_{x} × L_{y} × L_{z} = 16h × 8h × 4h, while the grid consisted of 512 × 256 × 128 nodes. In the results presented hereafter the scale h is used for normalizing lengths.
The vertical size of domain L_{z} = 4h was chosen quite small compared with the ones used in some numerical experiments (see, e.g., Grylls et al. 2020, and references therein). We adopted this compromise in order to unify LES runs for different stratification and at the same time avoid the significant computational overhead that grows with increasing stability. This was discussed in Xie et al. (2013), where the same value of L_{z} = 4h was used in neutral or stable conditions, and it was shown that the domain height is less of a concern in these cases than under unstable stratification. We also relied on our previous experience (see Glazunov 2014a,c,b), where different L_{z} domain heights (4h, 6h, and 8h) were considered and no substantial effect on the results was found.
d. The algorithm for setting up a numerical experiment with prescribed beforehand parameters
According to MOST, the statistics of the stratified turbulent flow over the canopy strongly depends on the Obukhov length L. We also assumed that the aerodynamic and heat transfer properties of the surface as a whole can be sensitive to the value of this length, if it is comparable with the height of the streamlined objects. To study these issues, it is required to carry out numerical experiments by varying the value of L over a wide range. To achieve this, we used the following algorithm.
The flow is maintained by a constant in time and space external force
The discussed setup is used in the LES model to obtain a statistically stationary state defined by two given external parameters: L_{fix} and
The described algorithm is implemented in order to optimize computational costs. Traditional strategy would require a successive series of simulations (starting with a neutral stratification) with a gradual decrease in the given value of T_{s} in each of them and using the results of the previous LES run as initial data for the new one. Considering that the dependence of the heat flux on surface temperature is not known in advance, it would be difficult to obtain the particular range of h/L_{fix} values. In addition, according to our experience, due to the interactions between the wind speed and the surface heat flux, such an algorithm accelerates the achievement of equilibrium states and prevents the flow from reaching some longlived regimes, in which only the lower part of the computational domain is turbulent and the upper part is laminar (see Van de Wiel et al. 2012). For the new algorithm we verified that after reaching the equilibrium state and determining the surface temperature according to the algorithm described above, the simulation can be continued with the obtained fixed value of T_{s}. Moreover, this does not affect the statistical characteristics of the steady flow.
e. The set of prescribed parameters of numerical experiments and some preliminary results
Four simulations were performed in each of EXP1, EXP2, and EXP3 groups: at nearneutral stratification and at the values h/L_{fix} = 0.5, h/L_{fix} = 1.0, and h/L_{fix} = 2.0. In addition, one simulation was run at h/L_{fix} = 3.0 in EXP3 (where the average height of objects is lower). The simulations under nearneutral stratification were carried out according to the same scheme as described above, except that the buoyancy term was neglected in the equation for the vertical velocity. The results for nearneutral stratification correspond to turbulent heat transport when L ≫ h (the influence of the buoyancy on the flow dynamics is negligible) or to the turbulent transport of passive scalar concentration by a neutrally stratified flow. To avoid introducing additional notations and for brevity, we will hereafter refer to this regime as “neutral stratification” and denote the corresponding results as h/L = 0. Note that even at large values of h/L_{fix} the flow remains turbulent both above the buildings and inside the layer 0 < z < h (see Fig. 1e, which shows the snapshot of temperature anomalies under the highest stability).
The experiments for neutral stratification were initialized with logarithmic velocity profiles obtained by preliminary estimates of z_{0}_{u} and D_{u}. A constant initial temperature was set in the entire domain. Additional random noise of small amplitude was superimposed on the initial conditions. The initial data for each of the subsequent runs, as the parameter h/L_{fix} increased, were given as instantaneous data of already performed ones.
All simulations were run for at least 30 dimensionless time units. Time averaging was carried out over final intervals
Figure 2a shows the normalized heat fluxes
The spatial averaging is performed only for the part of the computational domain filled with air. Since the heat fluxes are continuous with height, introducing part of the building area into the horizontal slice results in discontinuities at “roof” heights in the values of the fluxes averaged in this way. The normalized heat fluxes (dashed line) are the same for any stratification, but the momentum fluxes differ within the layer 0 < z < h, indicating that the form drag also depends on the value of h/L (this is discussed in more detail in the following sections).
The conditions described above correspond to the following dimensional characteristics of numerical experiments. The height of the “buildings” is h = 16 m. Dynamic and thermal roughness lengths of “walls,” “roofs,” and “ground,” corresponding to the given values of the coefficients
3. Simulation results
a. Nearneutral stratification
Figure 3 shows the mean profiles of the dimensionless velocity
Estimated roughness lengths z_{0}_{u} and z_{0}_{t} and displacement heights D_{u} and D_{t} under the neutral stratification. The inverse Stanton number
Although streamlined objects can strongly influence the local in space mean velocity and temperature profiles within the roughness layer, it can be seen that their spaceaveraged counterparts fit the logarithmic profiles quite well. Analogous results were first obtained and discussed in Cheng and Castro (2002), where the measurement data above more dense urban canopies were analyzed.
In all cases, close values of displacement heights for temperature and velocity were obtained, D_{t} ≈ D_{u}. This indicates approximately the same effect of the complex surface geometry on the velocity and temperature turbulent length scales in the external flow.
In all simulations, we found very small values of z_{0}_{t} and large inverse Stanton numbers. For comparison, the specified coefficient
According to the velocity profiles shown in Fig. 3 and the values given in Table 1, the canopy in EXP1 has the highest aerodynamic roughness, while the canopies in EXP2 and EXP3 provide approximately the same drag to the external flow. At the same time, the most efficient surface heat exchange occurs in EXP3, where the greatest values of thermal roughness length z_{0}_{t} and minimal inverse Stanton numbers are obtained. Thus, there is no unambiguous dependence of z_{0}_{t} on z_{0}_{u}: surface EXP1, which manifests itself as the surface that provides the greatest aerodynamic drag, results in a weakly effective scalar exchange, while the surfaces EXP2 and EXP3, which have similar aerodynamic characteristics, differ in heat exchange characteristics.
It should be noted that the expression (18) for volume drag is rather formal. In fact, the average force acting on the air at a certain height depends not only on the average velocity U_{air} at the same height, but also on the entire threedimensional flow structure. Volumetric drag coefficient C_{D}(z) specified by the formula (18) in instance may be undefined, or may have negative values (e.g., for the flow in the “urban canyon,” where the mean velocity changes sign near the ground). In the cases considered here, one can use Eq. (18) due to the low packing density, which allows us to consider representation of the canopy layer as some form of porous medium. A similar approach is widely used to construct singlecolumn multilayer RANS models of urban ABL (see, e.g., Nazarian et al. 2020, and references therein).
The dimensionless volumetric drag coefficients C_{D}h depending on the height are shown in Fig. 3d. The coefficients C_{D} in EXP2 and EXP3 practically coincide if z > h/2. For both of these geometries, the ratio
It is interesting to note that approximately the same values of z_{0}_{u} were obtained (see Table 1 and dimensionless velocity profiles in Fig. 3a) in EXP2 and EXP3, where the coefficient C_{D} values are close in the upper part of the canopy and at the same time differ significantly in the lower part. That is, the surfaces EXP2 and EXP3 are very close in terms of the average aerodynamic effect on the external flow. It can be assumed that the average frictional stress at the surfaces under consideration is mainly determined by the geometric characteristics of the upper part of the canopy. This assumption requires a separate verification and will be implicitly confirmed by the analysis of stably stratified flows presented in the following sections.
b. Stable stratification
1) Influence of stratification on heat and momentum air–surface exchange
Figure 4 shows the results of all 13 LES runs: the dimensionless velocity profiles
Under strong stability and at the large values of (z − D)/h, these profiles are close to linear and, therefore, both z_{0} and D parameters cannot be reliably determined in all cases. Note for clarity that decreasing the value of z_{0} shifts the profiles to the right, while changing the value of D shifts these profiles vertically. In EXP1, where the largest values of D_{u} and D_{t} were found, we have determined all values of D_{u}, D_{t}, z_{0}_{u}, and z_{0}_{t} for h/L = 0, h/L = 0.38, and h/L = 0.75. At h/L = 1.39, only the roughness lengths z_{0}_{u} and z_{0}_{t} were varied, while the displacement heights D_{u} and D_{t} were fixed at their values under the neutral stratification. In EXP2 and EXP3, the values of D_{u} and D_{t} were determined at h/L = 0 only and then fixed. In EXP3, where the displacement heights are the smallest, i.e., the top of canopy exhibits properties far from those of a solid wall, the greatest deviations of the LES profiles from the universal ones, (4), are observed.
Considering shortcomings of the method for determining displacement heights and roughness lengths, in this subsection we restrict ourselves to more reliably defined exchange characteristics of complex surfaces, namely, the inverse of the Stanton number
Figure 4 (right column) shows that the values of
Based on the shape of the velocity and temperature profiles (Fig. 4), we can conclude that the increase in the values of
Figures 5a and 5b show the coefficients C_{U} and C_{T} depending on the parameter 〈h〉/L. Here 〈h〉 is the average height of the objects (〈h〉 = h in EXP1 and EXP3; 〈h〉 = 0.75h in EXP2).
In Fig. 5a these coefficients are normalized using values obtained for neutral stratification:
Figure 5b shows the same dependence of C_{U} and C_{T} with stability, but using a different normalization:
From Fig. 5b it follows that the formulas (4) predict well the behavior of the coefficient C_{U}. Thus, the influence of stratification on the aerodynamic properties of an urban canopy can be neglected. It can be assumed that z_{0}_{u} = const and D_{u} = const for all of the canopy geometries and values of the h/L parameter considered. A similar result was obtained in the wind tunnel experiment (Uehara et al. 2000), where the insensitivity of the value z_{0}_{u} to stratification was noted for the flow over urbantype objects.
On the other hand, the coefficient C_{T} obtained from the LES data for 〈h〉/L > 0 drops below the values of
The qualitative changes in the heat transfer process at 〈h〉/L > 1 is clearly seen in Fig. 5c, which shows C_{U}/C_{T} ratio of momentum and heat exchange coefficients. This ratio increases under the strong stability, when the parameter 〈h〉/L reaches values above unity. In particular, the alue of C_{U}/C_{T} obtained in EXP3 at h/L = 1.4 is almost twice as high compared to its value for neutral stratification.
2) On the applicability of linear dimensionless universal gradients (2) for the condition of strong surface cooling
In the previous subsection (see Fig. 5b), the values of
In Fig. 6a open symbols show the same values of Ri_{B}, but with fixed values of ln(z_{0}_{u}/z_{0}_{t}), which were obtained under neutral stratification. Four out of the 13 cases considered fall into the region, where universal linear dimensionless gradients (2) are inconsistent with such values of the inverse Stanton number.
When calculating fluxes in surface schemes of atmospheric models, the applicability of approximations (2) is limited by the range of values
The results show that the decrease in thermal roughness is noticeable already at values of the bulk Richardson number well below
Among other things, the results presented in this and preceding sections show that it is difficult to achieve greater static stability of the turbulent flow over the canopy, e.g., larger values of h/L, by further decreasing the surface temperature. Indeed, the thermal roughness length z_{0}_{t} decrease almost exponentially for large values of Ri_{B}. This means that the heat flux in this case starts to show weak dependance on the temperature difference (T − T_{s}) and is mostly determined by the wind velocity and the length scale associated with the geometry of streamlined objects, which defines the minimal Obukhov scale for external flow. A more precise analysis based on additional LES runs is needed to refine this regime.
3) Features of the flow dynamics inside the canopy layer
Let us trace how the flow dynamics in the layer 0 < z < h changes with increasing stability. As evident from Fig. 4 (left column), at the highest values of h/L (black curves) the mean velocity near the ground is relatively small, besides these profiles differ in shape inside the canopy from the profiles at lower stability (colored curves). Apparently, there is some kind of physical mechanism that leads to such a restructuring of the entire flow and to its deceleration.
Figure 7 shows the profiles of drag coefficient C_{D} which is calculated according to Eq. (18) for all LES runs. In the interval D_{u}/h < z/h < 1 and in all cases the coefficient C_{D} weakly depends on the stratification, but changes significantly in the underlying layer. However, such a strong change in the drag in the lower part of the canopy does not affect the aerodynamic characteristics of an urbantype surface as a whole—as was previously shown the values z_{0}_{u} and D_{u} weakly depend on h/L. This confirms our assumptions that these parameters mainly depend on the geometry of objects and the flow dynamics in the upper part of the canopy.
Near the ground, the coefficient C_{D} increases significantly at large values of h/L. For example, in EXP1 and EXP3, where a noticeable weakening of heat transfer is observed (see Figs. 4 and 5), the values of C_{D} near the ground at h/L ≈ 1.4 are an order of magnitude greater than under neutral stratification, and in EXP2 at h/L ≈ 2.0 this coefficient increased by two orders of magnitude.
Additional weakening of the heat flux inside the canopy layer in LES, associated with an increase in volume drag, can be due to two reasons. On the one hand, an increase in the coefficient C_{D} reduces the mean nearsurface velocity and its fluctuations, which, in turn, leads to a decrease in the absolute value of the surface heat flux parameterized by the formula (12). On the other hand, at a lower mean velocity, the generation of turbulent kinetic energy reduces both due to a decrease in the vertical shear production and due to a decrease in the generation of vortices by the turbulent flow around objects. This leads to a decrease in turbulent diffusion, which is clearly seen in the temperature profiles (see Fig. 4), which, at higher values of h/L, have very large vertical gradients near the ground surface. Similar effect was first observed in wind tunnel experiment (Uehara et al. 2000), where the stable conditions at inflow caused weakening of the downward flow into the street canyon, which facilitated the formation of a much more stable stratification inside the urban layer.
For a more accurate understanding of the processes occurring inside the canopy layer at large values of h/L, a detailed analysis of the energy balance is required, which was not carried out in this work. Here we restrict ourselves to the presentation of the characteristic flow structure, which, in our opinion, is associated with a possible mechanism for the increase in volumetric drag under the stable stratification.
Figure 8 shows the timeaveraged velocity near the ground (at the first level of the computational grid) in EXP3 under neutral stratification (left) and under stable conditions (right; h/L = 1.4). The streamlines and arrows in the figures above indicate the direction of the flow, and the gray colored contours (bottom panel) shows the values of normalized horizontal velocity
The average velocity under neutral stratification and the velocity under conditions of strong stability differ both in absolute value and in the flow structure. For the stable stratification, the flow around objects occurs along “widely diverging” trajectories, and the size of the recirculation zones increases both in front of the objects and behind them. Figure 9 shows the instantaneous velocity obtained in the same LES runs. It can be seen that in both cases the flow is turbulent. At the same time, the characteristic differences between the two types of flow structure noted above are clearly visible. Thus, stable stratification affects the flow in a manner similar to an increase in the effective size of streamlined objects. This effect may be related to the suppression of vertical motions by stable stratification, resulting in preferably twodimensional flow around obstacles rather than threedimensional flow under neutral stratification with partial transfer of mass over the roof surfaces. Thus, the mass must flow around a wider horizontal area, which results in a loss of flow momentum and is seen as an increase in the drag coefficient. Another physical mechanism that may lead to increased spatial scales is that a stably stratified turbulent flow becomes more anisotropic, approaching the characteristics of a twodimensional fluid with inherent inverse energy cascades. Clarifying these assumptions requires additional analysis, which is beyond the scope of this article.
4. Conclusions
The paper results and the main conclusions may be summarized as follows:

A new algorithm for threedimensional nonstationary simulation of stratified turbulent flows over surfaces of complex shape is proposed. This algorithm is designed to obtain an equilibrium state for prescribed beforehand values of parameters which define the characteristics of external turbulent flow. The equilibrium state is achieved by varying the surface temperature over time. The approach is applicable to both LES models and direct numerical simulations (DNS) models and can significantly reduce the computational cost when studying air–surface turbulent heat transfer.

Largeeddy simulations of neutrally and stably stratified flows over urbantype surfaces were performed, confirming the conclusions of Glazunov (2014c,b) that the aerodynamic properties of such surfaces weakly depend on stratification. In doing so, we extended the range of values of the parameter 〈h〉/L (where 〈h〉 is the average height of the roughness elements and L is the Obukhov length scale) and performed numerical experiments, including those over a surface with sparsely spaced objects. The results show that, at least in the cases considered, the dynamic roughness parameter z_{0}_{u} is mainly determined by the geometric characteristics of the upper layer of the urban canopy, while the configuration of objects near the ground and the turbulence dynamics in the lower part of canopy layer have no significant effect on the momentum exchange between the surface as a whole and the external flow.

It is shown that stable stratification at large values of the parameter 〈h〉/L causes a more pronounced decrease in the efficiency of air–surface heat exchange than that predicted by MOST. Formally, this effect can be represented as a significant decrease in the thermal roughness length z_{0}_{t}.

It was found that under stable stratification, the volumetric drag coefficient C_{D} significantly increases near the ground inside the canopy, thus slowing down the mean flow. We attributed this effect to a restructuring of the turbulent flow dynamics and showed that at 〈h〉/L > 1 the timeaverage velocity patterns near the ground are significantly different from those observed under neutral conditions.

An increase in C_{D} values with increasing static stability further attenuates the air–surface heat exchange and seems to be an effective mechanism for limiting the absolute value of the negative heat flux. Note that such mechanism can be seen as a plausible scenario for maintaining turbulence at large values of the Richardson number. Indeed, due to this mechanism, the heat exchange coefficient C_{U}C_{T} decreases with increasing stability much faster than the momentum exchange coefficient C_{U}C_{U}, in which one can see an analogy with the effect of increasing the turbulent Prandtl number Pr_{t} = K_{m}/K_{h} (here, K_{m} and K_{h} are the turbulent viscosity and diffusivity coefficients, respectively) at the supercritical Ri values. This effect was considered and formalized by Zilitinkevich et al. (2008b, 2013); see Fig. 3 in Zilitinkevich et al. (2008b) versus Fig. 5d in this paper.
Finally, we note that, although this results were obtained for an idealized geometry of roughness elements similar to urban environments, we believe that many of the qualitative relations can be generalized to a broader class of surfaces for which the condition 〈h〉/L ∼ 1 is feasible. For example, a sparse forest, an alternation of vegetation types, a smallscale topography, or a ridging sea ice surface may well satisfy this condition during strong nighttime temperature inversions, especially at high latitudes.
Correct calculation of the surface heat flux and tangential frictional stress is crucial for the reliable performance of largescale numerical atmospheric models. There is a known problem inherent in most ABL parameterizations, which manifests itself as an excessive weakening of the interaction between the model atmosphere and the surface at large values of Ri_{B}. In particular, this effect is clearly seen as a frictional decoupling in numerical weather prediction models over urban areas (Jeričević and Grisogono 2006). This shortcoming can lead to large errors in nighttime nearsurface air temperature forecasts (Atlaskin and Vihma 2012; Battisti et al. 2017; Haiden et al. 2018; Esau et al. 2018, 2021). It forces developers of largescale models to introduce some artificial restrictions or use universal functions that have no threshold number Ri and are not in all cases physically and experimentally justified. We hope that the parameterization of presented effect of increasing inverse Stanton number with increasing bulk Richardson number will allow if not to improve the atmospheric models, in “tuning” of which a great experience is accumulated, then at least to introduce new and intuitive physical meaning to some of the imposed constraints in surface fluxes schemes.
Acknowledgments.
This paper is dedicated to the memory of Sergey Sergeevich Zilitinkevich. About two years ago we discussed this LES setup, and he even made some suggestions for building parameterizations based on future results. Unfortunately, we were too late and have no opportunity to discuss the findings. Numerical experiments under stable stratification and their analysis (section 3b) were performed with financial support of the Russian Science Foundation, Grant 217130023. LES model code development (section 2) was supported by the Moscow Center of Fundamental and Applied Mathematics at INM RAS (with the Ministry of Education and Science of the Russian Federation, Agreement 075152022286). The simulations of urban turbulence using parallel computing (section 3a) was partially supported by the Russian Ministry of Science and Higher Education, Agreement 075152021574.
Data availability statement.
The data presented in Fig. 4 (dimensionless velocity and dimensionless temperature difference) and the data shown in Fig. 7 (volumetric drag coefficient) are available at http://doi.org/10.23728/b2share.5ca9f5cb8de3428ba087eebd2e2a5f01. Additional LES results from these simulations are available from the authors on request.
REFERENCES
Atlaskin, E., and T. Vihma, 2012: Evaluation of NWP results for wintertime nocturnal boundarylayer temperatures over Europe and Finland. Quart. J. Roy. Meteor. Soc., 138, 1440–1451, https://doi.org/10.1002/qj.1885.
Basu, S., and F. PortéAgel, 2006: Largeeddy simulation of stably stratified atmospheric boundary layer turbulence: A scaledependent dynamic modeling approach. J. Atmos. Sci., 63, 2074–2091, https://doi.org/10.1175/JAS3734.1.
Basu, S., A. A. M. Holtslag, B. J. H. Van De Wiel, A. F. Moene, and G.J. Steeneveld, 2008: An inconvenient “truth” about using sensible heat flux as a surface boundary condition in models under stably stratified regimes. Acta Geophys., 56, 88–99, https://doi.org/10.2478/s116000070038y.
Battisti, A., O. C. Acevedo, F. D. Costa, F. S. Puhales, V. Anabor, and G. A. Degrazia, 2017: Evaluation of nocturnal temperature forecasts provided by the Weather Research and Forecast Model for different stability regimes and terrain characteristics. Bound.Layer Meteor., 162, 523–546, https://doi.org/10.1007/s105460160209y.
Beare, R. J., and Coauthors, 2006: An intercomparison of largeeddy simulations of the stable boundary layer. Bound.Layer Meteor., 118, 247–272, https://doi.org/10.1007/s1054600428206.
Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor. Climatol., 30, 327–341, https://doi.org/10.1175/15200450(1991)030<0327:FPOLSF>2.0.CO;2.
Boppana, V. B. L., Z.T. Xie, and I. P. Castro, 2014: Thermal stratification effects on flow over a generic urban canopy. Bound.Layer Meteor., 153, 141–162, https://doi.org/10.1007/s1054601499351.
Brutsaert, W., 2013: Evaporation into the Atmosphere: Theory, History and Applications. Vol. 1. Springer, 302 pp.
Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Fluxprofile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181–189, https://doi.org/10.1175/15200469(1971)028<0181:FPRITA>2.0.CO;2.
Chamberlain, A. C., 1966: Transport of gases to and from grass and grasslike surfaces. Proc. Roy. Soc. London, 290A, 236–265, http://doi.org/10.1098/rspa.1966.0047.
Cheng, H., and I. P. Castro, 2002: Near wall flow over urbanlike roughness. Bound.Layer Meteor., 104, 229–259, https://doi.org/10.1023/A:1016060103448.
Cheng, H., P. Hayden, A. G. Robins, and I. P. Castro, 2007: Flow over cube arrays of different packing densities. J. Wind Eng. Ind. Aerodyn., 95, 715–740, https://doi.org/10.1016/j.jweia.2007.01.004.
Cheng, W. C., and C.H. Liu, 2011: Largeeddy simulation of turbulent transports in urban street canyons in different thermal stabilities. J. Wind Eng. Ind. Aerodyn., 99, 434–442, https://doi.org/10.1016/j.jweia.2010.12.009.
Cheng, W. C., and F. PortéAgel, 2015: Adjustment of turbulent boundarylayer flow to idealized urban surfaces: A largeeddy simulation study. Bound.Layer Meteor., 155, 249–270, https://doi.org/10.1007/s1054601500041.
Cheng, W. C., and F. PortéAgel, 2021: A simple mixinglength model for urban canopy flows. Bound.Layer Meteor., 181, 1–9, https://doi.org/10.1007/s10546021006500.
Chenge, Y., and W. Brutsaert, 2005: Fluxprofile relationships for wind speed and temperature in the stable atmospheric boundary layer. Bound.Layer Meteor., 114, 519–538, https://doi.org/10.1007/s1054600414254.
Coceal, O., T. G. Thomas, I. P. Castro, and S. E. Belcher, 2006: Mean flow and turbulence statistics over groups of urbanlike cubical obstacles. Bound.Layer Meteor., 121, 491–519, https://doi.org/10.1007/s1054600690762.
Esau, I., M. Tolstykh, R. Fadeev, V. Shashkin, S. Makhnorylova, V. Miles, and V. Melnikov, 2018: Systematic errors in northern Eurasian shortterm weather forecasts induced by atmospheric boundary layer thickness. Environ. Res. Lett., 13, 125009, https://doi.org/10.1088/17489326/aaecfb.
Esau, I., and Coauthors, 2021: An enhanced integrated approach to knowledgeable highresolution environmental quality assessment. Environ. Sci. Policy, 122, 1–13, https://doi.org/10.1016/j.envsci.2021.03.020.
Germano, M., U. Piomelli, P. Moin, and W. H. Cabot, 1991: A dynamic subgridscale eddy viscosity model. Phys. Fluids, A3, 1760–1765, https://doi.org/10.1063/1.857955.
Glazunov, A. V., 2014a: Numerical modeling of turbulent flows over an urbantype surface: Computations for neutral stratification. Izv. Atmos. Oceanic Phys., 50, 134–142, https://doi.org/10.1134/S0001433814010034.
Glazunov, A. V., 2014b: Numerical simulation of stably stratified turbulent flows over an urban surface: Spectra and scales and parameterization of temperature and windvelocity profiles. Izv. Atmos. Oceanic Phys., 50, 356–368, https://doi.org/10.1134/S0001433814040148.
Glazunov, A. V., 2014c: Numerical simulation of stably stratified turbulent flows over flat and urban surfaces. Izv. Atmos. Oceanic Phys., 50, 236–245, https://doi.org/10.1134/S0001433814030037.
Glazunov, A. V., 2018: Numerical simulation of turbulence and transport of fine particulate impurities in street canyons. Numer. Methods Program., 19, 17–37, https://doi.org/10.26089/NumMet.v19r103.
Glazunov, A. V., Ü. Rannik, V. Stepanenko, V. Lykosov, M. Auvinen, T. Vesala, and I. Mammarella, 2016: Largeeddy simulation and stochastic modeling of Lagrangian particles for footprint determination in the stable boundary layer. Geosci. Model Dev., 9, 2925–2949, https://doi.org/10.5194/gmd929252016.
Glazunov, A. V., E. V. Mortikov, K. V. Barskov, E. V. Kadantsev, and S. S. Zilitinkevich, 2019: Layered structure of stably stratified turbulent shear flows. Izv. Atmos. Oceanic Phys., 55, 312–323, https://doi.org/10.1134/S0001433819040042.
Glazunov, A. V., A. V. Debolskiy, and E. V. Mortikov, 2021: Turbulent length scale for multilayer RANS model of urban canopy and its evaluation based on largeeddy simulations. Supercomput. Front. Innovation, 8, 100–116, https://doi.org/10.14529/jsfi210409.
Grachev, A. A., E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, 2012: Outlier problem in evaluating similarity functions in the stable atmospheric boundary layer. Bound.Layer Meteor., 144, 137–155, https://doi.org/10.1007/s1054601297149.
Grachev, A. A., E. L. Andreas, C. W. Fairall, P. S. Guest, and P. O. G. Persson, 2013: The critical Richardson number and limits of applicability of local similarity theory in the stable boundary layer. Bound.Layer Meteor., 147, 51–82, https://doi.org/10.1007/s1054601297710.
Gryanik, V. M., C. Lüpkes, A. Grachev, and D. Sidorenko, 2020: New modified and extended stability functions for the stable boundary layer based on SHEBA and parametrizations of bulk transfer coefficients for climate models. J. Atmos. Sci., 77, 2687–2716, https://doi.org/10.1175/JASD190255.1.
Grylls, T., I. Suter, and M. van Reeuwijk, 2020: Steadystate largeeddy simulations of convective and stable urban boundary layers. Bound.Layer Meteor., 175, 309–341, https://doi.org/10.1007/s1054602000508x.
Gryning, S.E., E. Batchvarova, and H. A. R. De Bruin, 2001: Energy balance of a sparse coniferous highlatitude forest under winter conditions. Bound.Layer Meteor., 99, 465–488, https://doi.org/10.1023/A:1018939329915.
Haiden, T., I. Sandu, G. Balsamo, G. Arduini, and A. Beljaars, 2018: Addressing biases in nearsurface forecasts. ECMWF Newsletter, No. 157, ECMWF, Reading, United Kingdom, 20–25, https://www.ecmwf.int/sites/default/files/elibrary/2018/18878addressingbiasesnearsurfaceforecasts.pdf.
Jeričević, A., and B. Grisogono, 2006: The critical bulk Richardson number in urban areas: Verification and application in a numerical weather prediction model. Tellus, 58A, 19–27, https://doi.org/10.1111/j.16000870.2006.00153.x.
Joffre, S. M., M. Kangas, M. Heikinheimo, and S. A. Kitaigorodskii, 2001: Variability of the stable and unstable atmospheric boundarylayer height and its scales over a boreal forest. Bound.Layer Meteor., 99, 429–450, https://doi.org/10.1023/A:1018956525605.
Kanda, M., M. Kanega, T. Kawai, R. Moriwaki, and H. Sugawara, 2007: Roughness lengths for momentum and heat derived from outdoor urban scale models. J. Appl. Meteor. Climatol., 46, 1067–1079, https://doi.org/10.1175/JAM2500.1.
Kazakov, A. L., and V. N. Lykosov, 1982: On the parameterization of the atmosphere interaction with the underlying surface in numerical modelling of atmospheric processes (in Russian). Proc. West Siberian Reg. Sci. Res. Hydrometeor. Inst., 55, 3–22.
Li, Z., T. Ming, S. Liu, C. Peng, R. de Richter, W. Li, H. Zhang, and C.Y. Wen, 2021: Review on pollutant dispersion in urban areas—Part A: Effects of mechanical factors and urban morphology. Build. Environ., 190, 107534, https://doi.org/10.1016/j.buildenv.2020.107534.
Louis, J.F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.Layer Meteor., 17, 187–202, https://doi.org/10.1007/BF00117978.
Marucci, D., and M. Carpentieri, 2019: Effect of local and upwind stratification on flow and dispersion inside and above a bidimensional street canyon. Build. Environ., 156, 74–88, https://doi.org/10.1016/j.buildenv.2019.04.013.
Massman, W. J., 1999: A model study of kBH−1 for vegetated surfaces using ‘localized nearfield′ Lagrangian theory. J. Hydrol., 223, 27–43, https://doi.org/10.1016/S00221694(99)001043.
Monin, A. S., and A. M. Obukhov, 1954: Basic laws of turbulent mixing in the surface layer of the atmosphere. Contrib. Geophys. Inst. Acad. Sci. USSR, 151, 163–187.
Morinishi, Y., T. S. Lund, O. V. Vasilyev, and P. Moin, 1998: Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys., 143, 90–124, https://doi.org/10.1006/jcph.1998.5962.
Nazarian, N., E. S. Krayenhoff, and A. Martilli, 2020: A onedimensional model of turbulent flow through “urban” canopies (MLUCM v2. 0): Updates based on largeeddy simulation. Geosci. Model Dev., 13, 937–953, https://doi.org/10.5194/gmd139372020.
Nieuwstadt, F. T. M., 1984: The turbulent structure of the stable, nocturnal boundary layer. J. Atmos. Sci., 41, 2202–2216, https://doi.org/10.1175/15200469(1984)041<2202:TTSOTS>2.0.CO;2.
Pahlow, M., M. B. Parlange, and F. PortéAgel, 2001: On Monin–Obukhov similarity in the stable atmospheric boundary layer. Bound.Layer Meteor., 99, 225–248, https://doi.org/10.1023/A:1018909000098.
Poggi, D., G. G. Katul, and J. D. Albertson, 2004: A note on the contribution of dispersive fluxes to momentum transfer within canopies. Bound.Layer Meteor., 111, 615–621, https://doi.org/10.1023/B:BOUN.0000016563.76874.47.
Rigden, A., D. Li, and G. Salvucci, 2018: Dependence of thermal roughness length on friction velocity across land cover types: A synthesis analysis using AmeriFlux data. Agric. For. Meteor., 249, 512–519, https://doi.org/10.1016/j.agrformet.2017.06.003.
Shen, Z., G. Cui, and Z. Zhang, 2017: Turbulent dispersion of pollutants in urbantype canopies under stable stratification conditions. Atmos. Environ., 156, 1–14, https://doi.org/10.1016/j.atmosenv.2017.02.017.
Tomas, J. M., M. J. B. M. Pourquie, and H. J. J. Jonker, 2016: Stable stratification effects on flow and pollutant dispersion in boundary layers entering a generic urban environment. Bound.Layer Meteor., 159, 221–239, https://doi.org/10.1007/s1054601501247.
Uehara, K., S. Murakami, S. Oikawa, and S. Wakamatsu, 2000: Wind tunnel experiments on how thermal stratification affects flow in and above urban street canyons. Atmos. Environ., 34, 1553–1562, https://doi.org/10.1016/S13522310(99)004100.
van der Linden, S. J. A., and Coauthors, 2019: Largeeddy simulations of the steady wintertime Antarctic boundary layer. Bound.Layer Meteor., 173, 165–192, https://doi.org/10.1007/s10546019004614.
Van de Wiel, B. J. H., A. F. Moene, and H. J. J. Jonker, 2012: The cessation of continuous turbulence as precursor of the very stable nocturnal boundary layer. J. Atmos. Sci., 69, 3097–3115, https://doi.org/10.1175/JASD12064.1.
Verhoef, A., H. A. R. De Bruin, and B. J. J. M. Van Den Hurk, 1997: Some practical notes on the parameter kB−1 for sparse vegetation. J. Appl. Meteor. Climatol., 36, 560–572, https://doi.org/10.1175/15200450(1997)036<0560:SPNOTP>2.0.CO;2.
Wyngaard, J. C., and O. R. Coté, 1972: Cospectral similarity in the atmospheric surface layer. Quart. J. Roy. Meteor. Soc., 98, 590–603, https://doi.org/10.1002/qj.49709841708.
Xie, Z., and I. P. Castro, 2006: LES and RANS for turbulent flow over arrays of wallmounted obstacles. Flow Turbul. Combust., 76, 291–312, https://doi.org/10.1007/s1049400690186.
Xie, Z., P. Hayden, and C. R. Wood, 2013: Largeeddy simulation of approachingflow stratification on dispersion over arrays of buildings. Atmos. Environ., 71, 64–74, https://doi.org/10.1016/j.atmosenv.2013.01.054.
Yang, R., and M. A. Friedl, 2003: Determination of roughness lengths for heat and momentum over boreal forests. Bound.Layer Meteor., 107, 581–603, https://doi.org/10.1023/A:1022880530523.
Zilitinkevich, S., 1995: Nonlocal turbulent transport: Pollution dispersion aspects of coherent structure of connective flows. WIT Trans. Ecol. Environ., 9, 53–60, https://doi.org/10.2495/AIR950071.
Zilitinkevich, S., I. Mammarella, A. A. Baklanov, and S. M. Joffre, 2008a: The effect of stratification on the aerodynamic roughness length and displacement height. Bound.Layer Meteor., 129, 179–190, https://doi.org/10.1007/s1054600893079.
Zilitinkevich, S., T. Elperin, N. Kleeorin, I. Rogachevskii, I. Esau, T. Mauritsen, and M. W. Miles, 2008b: Turbulence energetics in stably stratified geophysical flows: Strong and weak mixing regimes. Quart. J. Roy. Meteor. Soc., 134, 793–799, https://doi.org/10.1002/qj.264.
Zilitinkevich, S., T. Elperin, N. Kleeorin, I. Rogachevskii, and I. Esau, 2013: A hierarchy of energyand fluxbudget (EFB) turbulence closure models for stablystratified geophysical flows. Bound.Layer Meteor., 146, 341–373, https://doi.org/10.1007/s1054601297688.