Studies of Stable Stratification Effect on Dynamic and Thermal Roughness Lengths of Urban-Type Canopy Using Large-Eddy Simulation

Andrey Glazunov aMarchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia
bResearch Computing Center, Lomonosov Moscow State University, Moscow, Russia
dMoscow Center for Fundamental and Applied Mathematics, Moscow, Russia

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Evgeny Mortikov bResearch Computing Center, Lomonosov Moscow State University, Moscow, Russia
aMarchuk Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia

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Andrey Debolskiy bResearch Computing Center, Lomonosov Moscow State University, Moscow, Russia
cA. M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia

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Abstract

Large-eddy simulations (LES) of neutrally and stably stratified turbulent flows over urban-type surfaces with relatively low plan area ratios are presented. Numerical experiments were performed for different shapes of streamlined objects and at different static stability. A new method for setting up a numerical experiment aimed at studying the heat and momentum transfer within the roughness layer and investigating the thermal and dynamic interaction between the turbulent flow and the surface as a whole has been proposed. This method enables us to obtain an equilibrium state for values of parameters determining the characteristics of the external turbulent flow chosen beforehand. A strong dependence of the thermal roughness length on stratification was found. We also discuss the physical mechanisms that lead to the maintenance of turbulence above the canopy when the ground surface is strongly cooled.

Significance Statement

Using LES, we identify a mechanism that contributes to the maintenance of turbulence in the atmospheric boundary layer under the condition of strong surface cooling. Although these results are obtained for an urban canopy, we believe that the qualitative conclusions should be general for a wide type of surfaces with large-scale roughness elements. We hope that the new results will be useful for improving surface flux schemes in NWP and climate atmospheric models that suffer from attenuated mixing in a very stable boundary layer and the effect of “surface decoupling.” The found effect gives a physically justified alternative way to parameterize the air–surface exchange under strong stability compared to the often ad hoc modification of the MOST universal functions.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Andrey Glazunov, glas@gmail.com

Abstract

Large-eddy simulations (LES) of neutrally and stably stratified turbulent flows over urban-type surfaces with relatively low plan area ratios are presented. Numerical experiments were performed for different shapes of streamlined objects and at different static stability. A new method for setting up a numerical experiment aimed at studying the heat and momentum transfer within the roughness layer and investigating the thermal and dynamic interaction between the turbulent flow and the surface as a whole has been proposed. This method enables us to obtain an equilibrium state for values of parameters determining the characteristics of the external turbulent flow chosen beforehand. A strong dependence of the thermal roughness length on stratification was found. We also discuss the physical mechanisms that lead to the maintenance of turbulence above the canopy when the ground surface is strongly cooled.

Significance Statement

Using LES, we identify a mechanism that contributes to the maintenance of turbulence in the atmospheric boundary layer under the condition of strong surface cooling. Although these results are obtained for an urban canopy, we believe that the qualitative conclusions should be general for a wide type of surfaces with large-scale roughness elements. We hope that the new results will be useful for improving surface flux schemes in NWP and climate atmospheric models that suffer from attenuated mixing in a very stable boundary layer and the effect of “surface decoupling.” The found effect gives a physically justified alternative way to parameterize the air–surface exchange under strong stability compared to the often ad hoc modification of the MOST universal functions.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Andrey Glazunov, glas@gmail.com

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 urban-type 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.

Let us introduce the basic concepts and notations necessary for further presentation. As a rule, parameterizations of the interaction between the atmosphere and the underlying surface are based on the Monin–Obukhov similarity theory (MOST; Monin and Obukhov 1954), which establishes, in particular, the relations connecting the mean gradients and turbulent fluxes. According to MOST, the dimensionless gradients Φm and Φh for velocities and conservative scalars are universal functions of a single parameter ξ = z/L, where z is height and L is Obukhov length scale defined by
L=U*3FB,
where U*=|τs|1/2 is the friction velocity (τs is the surface frictional stress normalized to the air density), and FB is the buoyancy flux at the surface. In the case of dry atmosphere, FB = gQs/T0, where Qs = 〈Tw′〉 is the kinematic turbulent potential temperature flux inside the “constant flux layer” and at such distance from the surface that the molecular heat transfer is negligible; T0 is the characteristic absolute temperature value; g is the acceleration due to gravity. Note that the definition of the Obukhov length scale L used here and hereafter does not include the von Kármán constant κ. Further, for brevity, we will call the second moments 〈Tw′〉 and 〈uw′〉 heat and momentum fluxes, respectively, and refer to the potential temperature as the temperature. The turbulent temperature scale is introduced as T*=|Qs|/U*.
For stable stratification, the universal functions Φm(ξ) and Φh(ξ) are well approximated by linear relationships:
Φm(ξ)dUdzzU*=1κ+cmξ,Φh(ξ)dTdzzT*=Pr0κ+chξ,
where U and T are mean velocity and temperature, Pr0 is the turbulent Prandtl number at neutral stratification, cm and ch are universal constants. When approximating the average temperature and velocity profiles, we will use the following set of constants: κ = 0.4, cm = 5.0, ch = 4.0, and Pr0 = 0.8. These values are close to those obtained from measurements (e.g., Businger et al. 1971; Beljaars and Holtslag 1991; Pahlow et al. 2001; Grachev et al. 2012, 2013) and from LES and DNS data (e.g., Beare et al. 2006; Basu and Porté-Agel 2006; Glazunov 2014c; Glazunov et al. 2019). The effects typical of strong stability which may lead to nonlinear dimensionless gradients (see, e.g., Grachev et al. 2012, 2013; Glazunov 2014c) or an increase in the turbulent Prandtl number (see Zilitinkevich et al. 2013; Glazunov et al. 2019) are not considered here. In addition, we do not introduce local scaling (see Nieuwstadt 1984; Wyngaard and Coté 1972) because we will investigate more substantial and coarser effects [see the differences in “local” and “global” scaling obtained for similar turbulent flows in Glazunov (2014c)].

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′ = zD, 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〉).

By integrating the gradients over height, one can obtain so-called bulk formulas for surface heat and momentum fluxes:
|τs|=CU2U2,Qs=CUCTU(TTs),
where Ts is the surface temperature, and CU2 and CUCT are the height-dependent momentum and heat transfer coefficients. For the linear dimensionless gradients in (2), we have
CU=(UU*)1=(1κlnzDuz0u+cmzDuL)1,CT=(TTsT*)1=(Pr0κlnzDtz0t+chzDtL)1,
where z0u and z0t are the dynamic and thermal roughness lengths, respectively. Furthermore, we will also refer to Eq. (4) as approximations of the dimensionless velocity U/U* and dimensionless temperature difference (TTs)/T*. At values of Lz, i.e., under nearly neutral stratification, Eq. (4) approach logarithmic profiles for velocity and temperature. We introduced two different displacement heights Du and Dt, taking into account that the effect of surface geometry on the turbulent length scales for velocity and temperature fluctuations may be different. The dimensional parameters z0u and z0t appear in Eq. (4) because universal form of gradients (2) is not valid at heights z ∼ 〈h〉 and less. These parameters characterize the surface as a whole in terms of momentum and heat transfer efficiency.

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 z0u is determined only by the geometric properties of the surface and does not depend on Re. The value of z0u can be obtained from measurement data or numerical simulation results. Then, different surfaces are classified by their shape, and the value of z0u 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 urban-like canopies considered below, z0u/〈h〉 ∼ 10−2.

The greater difficulty lies in the estimation of the thermal roughness length z0t. There is no form drag for the heat flux, which implies z0tz0u, so that the heat exchange occurs, ultimately, due to the molecular heat transfer in a thin near-surface layer on the complex surface of streamlined objects. Accordingly, z0t depends not only on the geometry of the roughness elements, but also on the friction velocity U*, as well as on the coefficients of the kinematic molecular viscosity ν and thermal diffusivity α.

With almost neutral stratification, it is customary to compare the efficiency of turbulent air–surface heat exchange with the efficiency of momentum exchange through the inverse Stanton number:
κBh1=ln(z0u/z0t).
The Stanton number is expressed in this form if the turbulent Prandtl number (Pr0) is set equal to unity and z/L ≪ 1 (see, e.g., Verhoef et al. 1997). Taking into account stratification corrections and the fact that Pr0 ≠ 1, it will also be convenient to use the definition of κBh1 introduced by Chamberlain (1966), which in our notation will be written as follows:
κBh1=κ(|TTs|T*UU*)=κ(CUCTCUCT).
Note that the value of κBh1, defined by (6), generally depends on z, i.e., on the thickness of the air layer through which the turbulent transport occurs.
A variety of ways of expressing the number κBh1 or length z0t in terms of known external flow parameters and canopy characteristics have been proposed, ranging from simple relations based on dimensional analysis (e.g., Brutsaert 2013; Kanda et al. 2007; Zilitinkevich 1995; Yang and Friedl 2003) to complex models of heat transport in the vegetation layer (e.g., Massman 1999). A common feature of such parameterizations is that z0t decreases (κBh1 increases) with increasing roughness Reynolds number, Re*=U*z0u/ν. For example, Zilitinkevich (1995) suggests the following expression:
z0t=z0uexp(0.8Re*).
As was shown in Rigden et al. (2018), none of these parameterizations provide a universal fit to the measurement data obtained for different types of surfaces. At the same time, statistically significant diurnal variations of κBh1 values were recorded (see Rigden et al. 2018, Fig. 1), which may indicate a strong dependence on stratification. Such a dependence may not be excluded for values of L approximately equal to and less than the thickness of the layer h filled with roughness elements. That is, when stratification directly and significantly affects the dynamics of turbulent flow within the canopy layer. Indeed, the scale L was originally introduced by Monin and Obukhov (1954) as a height of the near-surface layer in which the influence of the atmospheric stratification is small and the turbulence is determined only by dynamic factors. If the value of L is small enough compared to the h, then the turbulence statistics inside the canopy would be different from that of a neutrally stratified flow.

In Zilitinkevich et al. (2008a) the dependencies of the dynamic roughness length z0u and displacement height Du on the parameter 〈h〉/L were proposed. According to Zilitinkevich et al. (2008a), z0u decreases, while Du 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 z0u and Du 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 time-stationary 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 urban-like 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 Porté-Agel 2015). This, in particular, allows LES data to be used to develop and to refine multilayer one-dimensional RANS models of urban ABL (Nazarian et al. 2020; Cheng and Porté-Agel 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 steady-state urban ABLs are the complexity of setting up a numerical experiment to simulate quasi-equilibrium 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 urban-like 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 urban-type 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 urban-like ABL at values Lh 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 z0t because the surface heat flux Qs rather than the surface temperature Ts 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 Ts with time. In this case, the heat flux Qs, which depends on the local instantaneous values of the near-surface air velocity uw and temperature Tw, is calculated. This allows one to pose the problem of identifying the relation between the thermal roughness length z0t 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

The LES model is based on a system of equations for an incompressible fluid and utilizes the Boussinesq approximation. This model explicitly reproduces the spatially filtered velocity u¯ and is supplemented by the filtered temperature T¯ transport equation. In tensor notation the equations have the following form:
u¯it+u¯iu¯jxj=τijxjp¯xi+δi3gT0T¯+Fiu,
u¯ixi=0,
T¯t+u¯iT¯xi=ϑitxi+Ft,
where Fiu corresponds to the external forces acting on the flow; Ft are bulk sources for the temperature; p¯ is the normalized pressure; τij=uiuj¯u¯iu¯j denotes the modeled subgrid/subfilter stress tensor; ϑit=uiT¯u¯iT¯ are the subgrid/subfilter heat fluxes; T0 is the reference temperature; the buoyancy parameter was fixed, g/T0 = 1/30 m s−2 K−1. Terms that include the molecular kinematic viscosity of air and thermal conductivity are neglected, as is customary in LES models of the atmospheric boundary layer.

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 energy-conservative 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 urban-type surfaces and has shown the ability to correctly reproduce some integral characteristics of such flows, e.g., the displacement height Du and roughness length z0u of the staggered cubes array (Glazunov 2014a) and the mean characteristics of the scalar turbulent transport inside the street-canyons 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 real-scale urban turbulence. In particular, the thin near-wall 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 UwO(1) m s−1 and the corresponding characteristic values of the near-wall friction velocity Uw*O(0.1)ms1, the thicknesses of the viscous sublayers yν will be on the order of 1 mm, which is easy to see from the estimate of the dimensionless thickness of this sublayer using the generally accepted formula: yν+=yνUw*/ν5. Accordingly, using LES or DNS, it is not possible to verify the relationship in (7) or other similar parameterizations at Re* values characteristic of real-scale urban turbulence.

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 CU and CT is valid only for flows with horizontal homogeneity. Therefore, we fix the values CUw,CTw=const (the index w means that these coefficients refer to a small fixed distance from the nearest surface). The following assumptions are introduced: (i) near any wall, the flow is turbulent at the “subgrid” level, and the walls themselves can be considered as a rough surface with Re*1; accordingly, the momentum and heat transfer laws have a quadratic form (3); (ii) the effect of stratification on the coefficients CUw and CTw is negligible (due to the proximity of the walls); (iii) the possible spatial distribution of CUw and CTw values over the surfaces has no crucial effect on the dynamics of large-scale turbulence inside the canopy layer and in the external flow above it.

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 z0t and inverse Stanton number κBh1 may differ from the values of these parameters in a real urban environment or in a laboratory experiment performed in similar conditions. However, we expect that the trends of their values due to the influence of stratification will be correctly reproduced by the LES model.

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 urban-like 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 λp, 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 C20S-6.25% of wind tunnel measurements in Cheng et al. (2007).

Fig. 1.
Fig. 1.

Scheme illustrating the setup of numerical experiments: (a)–(c) configuration of streamlined objects, vertical heights of elements is indicated next to the surface of the blocks; (d) schematic representation of boundary conditions and external forcing; (e) visualization of the instantaneous state of temperature anomalies in EXP3 at h/Lfix = 2.0 [a fragment of the area shown in (c) by a dotted line]; the isolines and color on the cross sections show the normalized temperature difference (T¯Ts)/T*; isosurface is shown at (T¯Ts)/T*=80.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

The size of the entire computational domain was Lx × Ly × Lz = 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 Lz = 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 Lz = 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 Lz domain heights (4h, 6h, and 8h) were considered and no substantial effect on the results was found.

Periodic boundary conditions were applied in horizontal directions. The free slip conditions for the velocity and zero heat flux were set at the top boundary of domain. Frictional stresses were calculated on the lower boundary of the domain and on all surfaces of objects as follows:
τw=CUwCUw|uw|uw,
where uw is the instantaneous velocity tangent to the corresponding surface at a distance Δ/2 from it. The value of the momentum exchange coefficient was set the same for all surfaces with CUw=0.124.
The heat flux on the “walls” and “roofs” of “buildings” was set equal to zero, and on the surface of the “ground” it was calculated by the formula
Qs=CTwCUw|uw|(TwTs).
Here Tw is the instantaneous value of the air temperature at the distance Δ/2 from the surface, and CTw=CUw/Pr0=0.155. It should be noted that we restrict ourselves with fixing the value of the CTw coefficient, despite the potential dependence of the near-wall heat transfer on Re* values. In the case of a flat surface, this corresponds to prescribing a fixed value of the thermal roughness parameter z0tw, which is still common practice in many SBL simulations (see, e.g., Beare et al. 2006; van der Linden et al. 2019).

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.

It was assumed that the surface temperature Ts does not depend on spatial coordinates, but depends on time and is calculated so that the following relation is satisfied:
1LxLygT0SQsdS=1Lfix(|uw|h)3/2,
where S is the surface of the of the “ground” area; Lfix = const is the prescribed value of the Obukhov length scale; |〈uw′〉|h is the momentum flux at height z = h + 1.5Δ, averaged over the horizontal plane (dashed horizontal line in schematic Fig. 1d). The flux |〈uw′〉|h includes both the resolved and subgrid/subfilter parts. It is computed at each time step of the LES model. Using the relationships (13) and (12), we have
Ts=[S|uw|TwdST0gLxLyCTwCUwLfix(|uw|h)3/2]/S|uw|dS.
An additional positive term is added to the right-hand side of Eq. (10), which describes the volume heating and does not depend on the spatial coordinates:
Ft=1VairSQsdS,
where Vair is the volume of the part of the computational domain filled with air. Such heating provides a zero temporal trend of the temperature averaged over the entire domain volume. In the Boussinesq approximation, such heating does not change the buoyancy gradients and, therefore, does not affect the evolution of velocity and temperature anomalies.

The flow is maintained by a constant in time and space external force Fu=(F1u,0,0) directed along X axis. The value of this force is defined as F1u=U*fix2/(Lzh), where U*fix is the prescribed value of the friction velocity equal to the expected value |〈uw′〉|1/2 at height h in equilibrium.

The discussed setup is used in the LES model to obtain a statistically stationary state defined by two given external parameters: Lfix and U*fix. Simulations show that such quasi equilibrium can be achieved in a fairly wide range of values of the stability parameter h/L*fix. In particular, this manifests itself in the linear (predetermined by the experimental setup) form of momentum and heat fluxes at heights z > h (see Fig. 2a), that are evident after some time t˜eqv1. In Fig. 2 the fluxes are averaged horizontally and across time intervals [t˜;t˜+Δ˜t], where t˜=tU*fix/Lz is the dimensionless time; Δ˜t is time averaging interval; t˜>t˜eqv ; Δ˜t1. The specific value of the dimensionless time to reach the quasi-equilibrium state t˜eqv and the value Δ˜t will be specified in the next subsection. Averaged over these time intervals surface temperature TsΔt also reaches a nearly constant value. Note that some temporal temperature variations Ts(t) will also be present at times t˜>t˜eqv due to the variability of the spatially averaged terms in the right-hand side of Eq. (14). The amplitude of these variations depends on the horizontal size of the domain and will decrease with its extension. By surface temperature Ts in what follows we will mean its time-averaged value.

Fig. 2.
Fig. 2.

(a) Dimensionless heat and momentum fluxes in EXP1 depending on stratification. The total fluxes are shown, including both the resolved and the subgrid/subfilter parts. (b) Height-dependent Obukhov lengths obtained from the fluxes at the corresponding altitudes z.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

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 Ts 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/Lfix 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 long-lived 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 Ts. 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 near-neutral stratification and at the values h/Lfix = 0.5, h/Lfix = 1.0, and h/Lfix = 2.0. In addition, one simulation was run at h/Lfix = 3.0 in EXP3 (where the average height of objects is lower). The simulations under near-neutral 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 near-neutral stratification correspond to turbulent heat transport when Lh (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/Lfix 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 z0u and Du. 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/Lfix 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 Δ˜t=10.

The results will be presented in dimensionless form using the friction velocity
U*=(|uwxyt|z=h+1.5Δ)1/2
and the temperature scale
T*=(|Twxyt|z=h+1.5Δ)/U*,
i.e., turbulent scales calculated from mean fluxes in the immediate vicinity of the object′s maximum height h (dashed horizontal line in Fig. 1d). The near-surface Obukhov length scale L is calculated from these values U* and T*, which turned out to be approximately 4/3 times greater than the specified values of Lfix due to the decrease in the absolute value of the heat flux with height in the layer 0 < z < h. Note that for insufficiently large values of the averaging window Δ˜t and the total integration time some deviations from the given values of U*fix and Lfix may occur. These deviations will be especially noticeable at large values of h/L, for which the adaptation time of the velocity and temperature profiles is rather long due to the small values of turbulent viscosity and diffusivity. Thus, normalization using the internal turbulent scales U* and T* is preferable for intercomparison of the dimensionless quantities analyzed below.

Figure 2a shows the normalized heat fluxes Twxyt/(U*T*) and momentum fluxes uwxyt/(U*2) as a functions of z/h depending on h/L in the state of quasi equilibrium. Here, the fluctuations in temperature and velocity correspond both to unsteady turbulence and time-averaged anomalies; therefore, these fluxes also include dispersive terms formed by stationary eddies.

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 CTw and CUw, are z0uw=0.01m and z0tw=0.01m. The friction velocity is U*U*fix=0.25ms1. The wind velocity U10 varies from 2.5 to 6 m s−1, and the temperature difference (T10Ts) lies in the interval ∼1°–15°C depending on the stratification and geometry of the surface (here T10 and U10 are temperature and velocity at a height of 10 m above the “roofs”). The numerical simulation time of each experiment is nearly two hours, and the averaging interval is about 45 min. The spatial grid step is set to Δ = 0.5 m, while the time step is chosen to ensure numerical stability and varies from 0.01 to 0.03 s, depending on the conditions of particular LES run.

3. Simulation results

a. Near-neutral stratification

Figure 3 shows the mean profiles of the dimensionless velocity U/U* (Fig. 3a) and the dimensionless temperature difference (TTs)/T* (Fig. 3b), obtained in EXP1, EXP2, and EXP3 under neutral stratification (at Lh). The dashed curves are the approximations by logarithmic dependencies (4). These approximations are estimated by minimizing their standard deviation from the corresponding LES profiles in the height interval [1.1h, 2.6h] while changing the values of z0u and Du or z0t and Dt. The obtained values of various parameters are given in Table 1. The choice of this height interval is not strictly justified; however, when it changes, the parameters of logarithmic dependency may also change slightly, but the general trends, considered bellow, remain. The dependence of these parameters on the choice of height interval for fitting and on the method for estimation of friction velocity was discussed in Cheng et al. (2007). The values of z0u and Du obtained from LES data in EXP1 (see Table 1) are close to the values of these parameters (z0u/h = 0.024 and Du/h = 0.704) obtained by Cheng et al. (2007), where direct measurements of the stress over the canopy were used to obtain the friction velocity U* and both parameters were calculated by extending the logarithmic region of the velocity profile (see Cheng et al. 2007, Table 2, penultimate row), which is the closest definition with respect to ours.

Fig. 3.
Fig. 3.

Neutral stratification. (a) Nondimensional mean velocity U/U* and (b) nondimensional mean temperature difference (TTs)/T* depending on the surface geometry [solid colored lines are the LES data, and dashed lines are log-linear approximations (4)]. (c) Normalized momentum and heat fluxes (averaging over the area filled with air). (d) Normalized volumetric drag coefficient hCD. (a) and (d) are reproduced using data from Glazunov et al. (2021).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

Table 1

Estimated roughness lengths z0u and z0t and displacement heights Du and Dt under the neutral stratification. The inverse Stanton number κBh1|10 is calculated using Eq. (6) at a height z/h = 1.64 (∼10 m above the “roofs”). The last two columns show the ratios of the exchange coefficients (at a height ∼10 m above the “roofs”) to their values above a flat surface with given values of roughness lengths z0uw=0.01m and z0tw=0.01m.

Table 1

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 space-averaged 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, DtDu. 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 z0t and large inverse Stanton numbers. For comparison, the specified coefficient CTw=0.155 corresponds to the value z0tw/h=6.25×104 (for a flat wall), that is, three to four orders of magnitude higher than the values z0t presented in Table 1. Thus, placing urban-type objects on the ground significantly reduces the coefficient CT even at a relatively low plan area density (e.g., in EXP3: Ssb/Sstot=0.015625, where Ssb is the area occupied by objects, and Sstot is the total area; that is, only about 1.6% of the surface is occupied by the “buildings”). This does not mean that at constant values of the external flow velocity U and temperature difference (TTs) the placement of such objects on the ground will reduce the absolute value of the heat flux |Qs| significantly. Indeed, while the values of z0t decrease, the values of z0u increase. The last two columns in Table 1 show the ratios of the momentum and heat exchange coefficients: (CU)2/(CUf)2 and (CUCT)/(CUfCTf). Here CU and CT are the coefficients obtained from the LES data at the height z = 1.64h, and CUf,CTf are the coefficients at the same height above the flat surface with prescribed roughness lengths z0tw/h=6.25×104 and z0ww/h=6.25×104; CUf and CTf are estimated using Eq. (4). Thus, after the placement of large objects (explicitly resolved in LES), the aerodynamic drag increases significantly, and the turbulent heat exchange between “air” and “ground” remains approximately the same.

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 z0t and minimal inverse Stanton numbers are obtained. Thus, there is no unambiguous dependence of z0t on z0u: 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.

Using the distribution of momentum fluxes (see Fig. 3c), one can estimate the force acting on a unit volume of air inside the canopy. Here we do not divide this force into two components, one of which is related to the tangential friction stress on the walls, and the other one related with the form drag due to the pressure difference between opposite sides of the objects. Assuming that at large Re values, both of these components are proportional to the square of the mean air velocity, the momentum balance in the equilibrium state can be written as follows:
CDUair2=Pxuwairz;
here the index “air” means that the averaging is carried out over the area occupied by air.

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 Uair at the same height, but also on the entire three-dimensional flow structure. Volumetric drag coefficient CD(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 single-column multilayer RANS models of urban ABL (see, e.g., Nazarian et al. 2020, and references therein).

The dimensionless volumetric drag coefficients CDh depending on the height are shown in Fig. 3d. The coefficients CD in EXP2 and EXP3 practically coincide if z > h/2. For both of these geometries, the ratio Sflayer/Vlayer=0.03125/h is the same in the upper half of the canopy, where Sflayer is the frontal area of the objects, and Vlayer is the volume of the layer. Apparently, it is this dimensional parameter that determines the value of CD, and the mutual arrangement of objects plays a secondary role when their ground location is quite sparse. In EXP1, where Sflayer/Vlayer at z > h/2 is twice as large as in EXP2 and EXP3, the coefficient CD also increases by about a factor of 2 in the upper half of the canopy. In all this cases, the coefficient CD grows strongly near the ground surface.

It is interesting to note that approximately the same values of z0u were obtained (see Table 1 and dimensionless velocity profiles in Fig. 3a) in EXP2 and EXP3, where the coefficient CD 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 U/U* (left column), the dimensionless temperature differences (TTs)/T* (center column), and the profiles of the inverse Stanton number κBh1 calculated according to (6) (right column). The frames are arranged in rows according to the surface configuration. Each colored curve corresponds to different value of stability parameter h/L. Approximations of velocity and temperature profiles by log-linear universal functions (4) are shown by dashed lines. The fitting method is similar to that described in section 3a.

Fig. 4.
Fig. 4.

(left) Nondimensional mean velocity U/U* and (center) mean temperature difference profiles (TTs)/T* Solid curves are the LES data. Dashed curves are the log-linear approximations (4). (right) Inverse Stanton number; see Eq. (6).

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

Under strong stability and at the large values of (zD)/h, these profiles are close to linear and, therefore, both z0 and D parameters cannot be reliably determined in all cases. Note for clarity that decreasing the value of z0 shifts the profiles to the right, while changing the value of D shifts these profiles vertically. In EXP1, where the largest values of Du and Dt were found, we have determined all values of Du, Dt, z0u, and z0t for h/L = 0, h/L = 0.38, and h/L = 0.75. At h/L = 1.39, only the roughness lengths z0u and z0t were varied, while the displacement heights Du and Dt were fixed at their values under the neutral stratification. In EXP2 and EXP3, the values of Du and Dt 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 κBh1, (6), and the coefficients CU and CT. Here, we will use the values of these coefficients at a height of z/h = 1.64, i.e., at a height of ∼10 m above the canopy in dimensional length units.

Figure 4 (right column) shows that the values of κBh1 increase with the growth of the parameter h/L. The inverse Stanton number reaches its highest values at h/L = 1.4 in EXP3, that is, for the canopy configuration, where the value of κBh1 is minimal under neutral stratification (see Table 1). Thus, there is no unambiguous relation between the Stanton number under neutral stratification and its values as stability increases.

Based on the shape of the velocity and temperature profiles (Fig. 4), we can conclude that the increase in the values of κBh1 with increasing stability occurs mainly due to a decrease in the thermal roughness lengths [profiles (TTs)/T* shift to the right as h/L increases, which corresponds to a decrease in z0t]. At the same time, no significant shifts of the velocity profiles are observed, but only their “slopes” increase due to growing of the dimensionless gradient Φm.

Figures 5a and 5b show the coefficients CU and CT 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).

Fig. 5.
Fig. 5.

Coefficients CU and CT (at height z/h = 1.64) depending on the geometry of the canopy and the parameter 〈h〉/L. (a) The coefficients are normalized to their values under neutral stratification. (b) The coefficients are normalized to the values of the coefficients obtained from Eq. (4) with the values z0u, z0t, Du, and Dt defined under neutral stratification. (c) Ratio CU/CT.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

In Fig. 5a these coefficients are normalized using values obtained for neutral stratification: C˜U=CU(h/L)/CU(0), C˜T=CT(h/L)/CT(0). Both CU and CT decrease with an increase in 〈h〉/L, which in part is ensured by the growth of the dimensionless gradients Φm and Φh.

Figure 5b shows the same dependence of CU and CT with stability, but using a different normalization: C˜U=CU(h/L)/CU0(h/L) and C˜T=CT(h/L)/CT0(h/L). Here CU0 and CT0 calculated using Eq. (4) for fixed values of parameters z0u, z0t, Du, and Dt defined at h/L = 0 (see Table 1).

From Fig. 5b it follows that the formulas (4) predict well the behavior of the coefficient CU. Thus, the influence of stratification on the aerodynamic properties of an urban canopy can be neglected. It can be assumed that z0u = const and Du = 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 z0u to stratification was noted for the flow over urban-type objects.

On the other hand, the coefficient CT obtained from the LES data for 〈h〉/L > 0 drops below the values of CT0. The difference becomes more pronounced with an increase in stability, e.g., in EXP3 the value of CT at h/L = 1.4 is approximately 3 times less than CT0. Since the dimensionless temperature gradients do not change significantly compared to the universal functions (2) (see the dashed lines in Fig. 4), this effect can be correctly taken into account in the formulas (4) only by significant reduction of the thermal roughness length z0t.

The qualitative changes in the heat transfer process at 〈h〉/L > 1 is clearly seen in Fig. 5c, which shows CU/CT 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 CU/CT 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 CU0 and CT0 used for normalization were calculated using the Obukhov length scale L obtained in LES and with the fixed values of the surface parameters z0u, z0t, Du and Dt. However, the system of Eqs. (3) and (4) may not have a physically correct solution for any given values of these parameters and for any set of values (TTs), U and z (see, e.g., Basu et al. 2008). It is known that the linear dimensionless gradients (2) make it possible to find the consistent solution if the bulk Richardson number RiB does not exceed some critical value RiBcr.

We will neglect small differences between displacement heights Du and Dt by setting D = (Du + Dt)/2 and define the bulk Richardson number (RiB) as follows:
RiB=gT0TTsU2z,
where z′ = zD. Then, the system of Eqs. (3) and (4) is reduced (see Kazakov and Lykosov 1982) to one quadratic equation:
aξ2+bξ+c=0,
where ξ = z′/L, and the coefficients a, b, and c are expressed as follows:
a=cm2RiBch,b=1κ[ln(zz0u)(2cmRiBPr0)Pr0ln(z0uz0t)],c=RiB[1κln(zz0u)]2.
In Fig. 6a, the colored region shows the values of ξ = z′/L depending on RiB and ln(z0u/z0t), obtained as physically consistent solution of the system of Eqs. (3) and (4). These values are calculated for z′ and z0u values obtained in EXP3 at L/h = 1.4. Critical curves correspond to
RiBcr=14Pr0[ln(z0uz0t)+ln(zz0u)]2cmln(z0uz0t)ln(zz0u)+[(cmch)/Pr0][ln(zz0u)]2.
These curves separate the areas of real and complex roots of Eq. (20) and are plotted with solid (for EXP3) and dashed lines (for EXP1 and EXP2). Black symbols show the values of ln(z0u/z0t) and RiB obtained in LES. These symbols correspond to different LES runs and are located from left to right and from bottom to top as h/L increases. All LES data fall within the domain with real roots of Eq. (20), i.e., RiBRiBcr. Note that the exact fulfillment of this inequality is not expected due to the imprecise method for estimating the roughness lengths and displacement heights and the assumption DuDt when estimating the critical curve.
Fig. 6.
Fig. 6.

(a) Inverse Stanton number ln(z0u/z0t) vs the bulk Richardson number RiB (19); black symbols are LES results; open symbols are the same values of RiB with fixed values of ln(z0u/z0t) obtained under the neutral stratification; critical curves (solid and dashed lines) separate the areas of real and complex roots of Eq. (20); physically correct z′/L values are shown in color. (b) Black line: the values of κz′/L on the critical curve (22) as a function of RiB; symbols: LES data. (c) Black line: flux Richardson number Rif on the critical curve (22); symbols: LES data. All results are shown at the height z/h = 1.64. The color field in (a) and the curves in (b), (c) are shown for the displacement height D = (Du + Dt)/2 and roughness length z0u obtained in EXP3.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

In Fig. 6a open symbols show the same values of RiB, but with fixed values of ln(z0u/z0t), 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 RiB<RiBcr. This is one of the advantages of using nonlinear functions Φm(ξ) and Φh(ξ) that do not have this restriction (see, e.g., Louis 1979; Chenge and Brutsaert 2005; Gryanik et al. 2020). However, based on the LES data it may be concluded that the integral universal functions (4) approximate the profiles over the canopy quite well (see Fig. 4). Therefore, their significant modification in this case is not needed. In addition, with the fixed values of roughness length z0t, one would have to introduce a significant nonlinearity of Φm(ξ) and Φh(ξ) even for ξ < 1 to obtain the values of CU and CT coinciding with those obtained in LES. It contradicts both observations and simulations. Besides, the choice of such hypothetical functions cannot be universal and depends on the canopy geometry.

The results show that the decrease in thermal roughness is noticeable already at values of the bulk Richardson number well below RiBcr and at relatively small values of the stability parameter ξ = z′/L. Recall that the Obukhov length L is defined here without including the von Kármán constant κ, so the ξ values shown in color are 2.5 times larger than those often used in meteorology [the stability parameter κξ as a function of RiB obtained from LES data and its estimated values along the critical curve (22), where these estimates have a maximum, are shown in Fig. 6b]. That is, this effect takes place in the subcritical regime for external turbulent flow. This is also confirmed by the relatively small flux Richardson number Rif=FB/[(dU/dz)τ], whose values obtained in LES are shown in Fig. 6c along with estimates of its maximum values on the critical curve (22), which are calculated as Rif=(1/Φm)ξ.

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 z0t decrease almost exponentially for large values of RiB. This means that the heat flux in this case starts to show weak dependance on the temperature difference (TTs) 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 CD which is calculated according to Eq. (18) for all LES runs. In the interval Du/h < z/h < 1 and in all cases the coefficient CD 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 urban-type surface as a whole—as was previously shown the values z0u and Du 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.

Fig. 7.
Fig. 7.

(top) Profiles of normalized volumetric drag coefficient CDh vs stability parameter h/L in EXP1, EXP2, and EXP3; the h/L values are marked in the legend by colors. (bottom) As in the top row, but with logarithmic axes.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

Near the ground, the coefficient CD 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 CD 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 CD reduces the mean near-surface 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 time-averaged 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 |u|/U*. The white curve shows the recirculation zones, corresponding to regions where the average longitudinal velocity is negative.

Fig. 8.
Fig. 8.

Time-averaged near-ground velocity (at the height z = Δ/2) in EXP3. (left) Neutral stratification; (right) the stable stratification at h/L = 1.4. (top) Streamlines/arrows show the direction of the flow. (bottom) The absolute value of normalized velocity; the white curve outlines the recirculation zones.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

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 two-dimensional flow around obstacles rather than three-dimensional 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 two-dimensional fluid with inherent inverse energy cascades. Clarifying these assumptions requires additional analysis, which is beyond the scope of this article.

Fig. 9.
Fig. 9.

As in Fig. 8, but for the instantaneous state of flow.

Citation: Journal of the Atmospheric Sciences 80, 1; 10.1175/JAS-D-22-0044.1

4. Conclusions

The paper results and the main conclusions may be summarized as follows:

  • A new algorithm for three-dimensional 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.

  • Large-eddy simulations of neutrally and stably stratified flows over urban-type 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 z0u 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 z0t.

  • It was found that under stable stratification, the volumetric drag coefficient CD 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 time-average velocity patterns near the ground are significantly different from those observed under neutral conditions.

  • An increase in CD 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 CUCT decreases with increasing stability much faster than the momentum exchange coefficient CUCU, in which one can see an analogy with the effect of increasing the turbulent Prandtl number Prt = Km/Kh (here, Km and Kh 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 small-scale 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 large-scale 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 RiB. 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 near-surface air temperature forecasts (Atlaskin and Vihma 2012; Battisti et al. 2017; Haiden et al. 2018; Esau et al. 2018, 2021). It forces developers of large-scale 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 21-71-30023. 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 075-15-2022-286). The simulations of urban turbulence using parallel computing (section 3a) was partially supported by the Russian Ministry of Science and Higher Education, Agreement 075-15-2021-574.

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

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Save
  • Atlaskin, E., and T. Vihma, 2012: Evaluation of NWP results for wintertime nocturnal boundary-layer temperatures over Europe and Finland. Quart. J. Roy. Meteor. Soc., 138, 14401451, https://doi.org/10.1002/qj.1885.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Basu, S., and F. Porté-Agel, 2006: Large-eddy simulation of stably stratified atmospheric boundary layer turbulence: A scale-dependent dynamic modeling approach. J. Atmos. Sci., 63, 20742091, https://doi.org/10.1175/JAS3734.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 8899, https://doi.org/10.2478/s11600-007-0038-y.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 523546, https://doi.org/10.1007/s10546-016-0209-y.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beare, R. J., and Coauthors, 2006: An intercomparison of large-eddy simulations of the stable boundary layer. Bound.-Layer Meteor., 118, 247272, https://doi.org/10.1007/s10546-004-2820-6.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Beljaars, A. C. M., and A. A. M. Holtslag, 1991: Flux parameterization over land surfaces for atmospheric models. J. Appl. Meteor. Climatol., 30, 327341, https://doi.org/10.1175/1520-0450(1991)030<0327:FPOLSF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 141162, https://doi.org/10.1007/s10546-014-9935-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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: Flux-profile relationships in the atmospheric surface layer. J. Atmos. Sci., 28, 181189, https://doi.org/10.1175/1520-0469(1971)028<0181:FPRITA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chamberlain, A. C., 1966: Transport of gases to and from grass and grass-like surfaces. Proc. Roy. Soc. London, 290A, 236265, http://doi.org/10.1098/rspa.1966.0047.

    • Search Google Scholar
    • Export Citation
  • Cheng, H., and I. P. Castro, 2002: Near wall flow over urban-like roughness. Bound.-Layer Meteor., 104, 229259, https://doi.org/10.1023/A:1016060103448.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 715740, https://doi.org/10.1016/j.jweia.2007.01.004.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cheng, W. C., and C.-H. Liu, 2011: Large-eddy simulation of turbulent transports in urban street canyons in different thermal stabilities. J. Wind Eng. Ind. Aerodyn., 99, 434442, https://doi.org/10.1016/j.jweia.2010.12.009.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cheng, W. C., and F. Porté-Agel, 2015: Adjustment of turbulent boundary-layer flow to idealized urban surfaces: A large-eddy simulation study. Bound.-Layer Meteor., 155, 249270, https://doi.org/10.1007/s10546-015-0004-1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Cheng, W. C., and F. Porté-Agel, 2021: A simple mixing-length model for urban canopy flows. Bound.-Layer Meteor., 181, 19, https://doi.org/10.1007/s10546-021-00650-0.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chenge, Y., and W. Brutsaert, 2005: Flux-profile relationships for wind speed and temperature in the stable atmospheric boundary layer. Bound.-Layer Meteor., 114, 519538, https://doi.org/10.1007/s10546-004-1425-4.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Coceal, O., T. G. Thomas, I. P. Castro, and S. E. Belcher, 2006: Mean flow and turbulence statistics over groups of urban-like cubical obstacles. Bound.-Layer Meteor., 121, 491519, https://doi.org/10.1007/s10546-006-9076-2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Esau, I., M. Tolstykh, R. Fadeev, V. Shashkin, S. Makhnorylova, V. Miles, and V. Melnikov, 2018: Systematic errors in northern Eurasian short-term weather forecasts induced by atmospheric boundary layer thickness. Environ. Res. Lett., 13, 125009, https://doi.org/10.1088/1748-9326/aaecfb.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Esau, I., and Coauthors, 2021: An enhanced integrated approach to knowledgeable high-resolution environmental quality assessment. Environ. Sci. Policy, 122, 113, https://doi.org/10.1016/j.envsci.2021.03.020.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Germano, M., U. Piomelli, P. Moin, and W. H. Cabot, 1991: A dynamic subgrid-scale eddy viscosity model. Phys. Fluids, A3, 17601765, https://doi.org/10.1063/1.857955.

    • Crossref
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