E − ε − 〈θ ²〉 turbulence closure model for an atmospheric boundary layer including the urban canopy

Abstract

A modified three-parameter model of turbulence for a thermally stratified atmospheric boundary layer (ABL) is presented. The model is based on tensor-invariant parameterizations for the pressure–strain and pressure–temperature correlations that are more complete than the parameterizations used in the Mellor–Yamada model of level 3.0. The turbulent momentum and heat fluxes are calculated with explicit algebraic models obtained with the aid of symbol algebra from the transport equations for momentum and heat fluxes in the approximation of weakly equilibrium turbulence. The turbulent transport of heat and momentum fluxes is assumed to be negligibly small in this approximation. The three-parameter $E - \varepsilon - {\left\langle {\theta ^{2} } \right\rangle }$ model of thermally stratified turbulence is employed to obtain closed-form algebraic expressions for the fluxes. A computational test of a 24-h ABL evolution is implemented for an idealized two-dimensional region. Comparison of the computed results with the available observational data and other numerical models shows that the proposed model is able to reproduce both the most important structural features of the turbulence in an urban canopy layer near the urbanized ABL surface and the effect of urban roughness on a global structure of the fields of wind and temperature over a city. The results of the computational test for the new model indicate that the motion of air in the urban canopy layer is strongly influenced by mechanical factors (buildings) and thermal stratification.

Appendix

This appendix presents the governing system of equations of a turbulent horizontally inhomogeneous thermally stratified ABL for the two-dimensional test under study. Explicit analytic expressions are given for the turbulent momentum and heat fluxes, and numerical values are presented for the constants of the modified three-parameter model of turbulence. Detailed expressions are presented for the parameterization of the effects of urbanized surface roughness.

1.1 Governing system of equations for the turbulent ABL

For flows in a planetary boundary layer, some approximations can be used in the governing equations. In Eq. 1a, the rotation term can be approximated with the expression:

$$ - 2\varepsilon _{{ijk}} \Omega _{j} U_{k} = f_{c} \varepsilon _{{ij3}} U_{j} $$

where the axes x, y, and z are directed eastward, northward, and upward, respectively, and f _c  = 2Ω sinφ is the Coriolis parameter with the angular velocity of the Earth’s rotation Ω and latitude φ. The buoyancy effects are taken into account in the Boussinesq approximation, and, for a two-dimensional flow, the system of Eqs. 1a and 1b is written as:

$$U_{x} + W_{z} = 0$$

(13a)

$$U_{t} + UU_{x} + WU_{z} = - \frac{1}{\rho }P_{x} - {\left\langle {wu} \right\rangle }_{z} + fV + \ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{u} $$

(13b)

$$V_{t} + UV_{x} + WV_{z} = - {\left\langle {wv} \right\rangle }_{z} - fU + \ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{v} $$

(13c)

$$W_{t} + UW_{x} + WW_{z} = - \frac{1}{{\rho _{0} }}P_{z} - {\left\langle {ww} \right\rangle }_{z} + \beta \ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }g$$

(13d)

$$\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{t} + U\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{x} + W\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{z} = - {\left\langle {u\theta } \right\rangle }_{x} - {\left\langle {w\theta } \right\rangle }_{z} + \ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{\theta } $$

(13e)

The dependent variables in Eqs. 13a–13e are the mean (time-averaged) flow velocities U, V, and W in the directions of the x, y, and z axes, respectively, the mean pressure P, and the mean departure of potential temperature $\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }$ from a reference temperature T ₀. The parametric quantities in Eqs. 13a–13e include the volumetric expansion rate of air β (3.53 × 10⁻³ K⁻¹) and the mean air density ρ ₀. The lowercase letters denote turbulent fluctuations of the corresponding quantities. The Reynolds turbulent stresses τ _ij and the turbulent heat flux vector h _j in Eqs. 13a–13e requires modeling. Explicit algebraic models for the Reynolds stresses and the turbulent heat flux are formulated in the next section.

1.2 Algebraic expressions for turbulent momentum and heat fluxes

From Eq. 9a, with consideration for $b_{{ij}} = {\left\langle {u_{i} u_{j} } \right\rangle } - \frac{2}{3}E\delta _{{ij}} ,$ and from Eq. 10a for the vector of turbulent heat flux h _i  = 〈u _i θ〉, the following implicit system of equations for the turbulent momentum and heat fluxes is written in the boundary layer approximation:

$${\left\langle {u^{2} } \right\rangle } = \frac{2}{3}E - \frac{\tau }{3}{\left( {4\alpha _{2} \frac{{\partial U}}{{\partial z}}{\left\langle {uw} \right\rangle } - 2\alpha _{2} \frac{{\partial V}}{{\partial z}}{\left\langle {vw} \right\rangle } + 2\alpha _{3} \beta g{\left\langle {w\theta } \right\rangle }} \right)}$$

(14a)

$${\left\langle {v^{2} } \right\rangle } = \frac{2}{3}E - \frac{\tau }{3}{\left( {4\alpha _{2} \frac{{\partial V}}{{\partial z}}{\left\langle {vw} \right\rangle } - 2\alpha _{2} \frac{{\partial U}}{{\partial z}}{\left\langle {uw} \right\rangle } + 2\alpha _{3} \beta g{\left\langle {w\theta } \right\rangle }} \right)}$$

(14b)

$${\left\langle {w^{2} } \right\rangle } = \frac{2}{3}E + \frac{\tau }{3}{\left( {2\alpha _{2} \frac{{\partial U}}{{\partial z}}{\left\langle {uw} \right\rangle } + 2\alpha _{2} \frac{{\partial V}}{{\partial z}}{\left\langle {vw} \right\rangle } + 4\alpha _{3} \beta g{\left\langle {w\theta } \right\rangle }} \right)}$$

(14c)

$${\left\langle {uw} \right\rangle } = - \frac{\tau }{2}\frac{{\partial U}}{{\partial z}}2\alpha _{2} {\left\langle {w^{2} } \right\rangle } + \alpha _{3} \tau \beta g{\left\langle {u\theta } \right\rangle }$$

(14d)

$${\left\langle {vw} \right\rangle } = - \frac{\tau }{2}\frac{{\partial V}}{{\partial z}}2\alpha _{2} {\left\langle {w^{2} } \right\rangle } + \alpha _{3} \tau \beta g{\left\langle {v\theta } \right\rangle }$$

(14e)

$${\left\langle {uv} \right\rangle } = - \tau \alpha _{2} {\left( {\frac{{\partial V}}{{\partial z}}{\left\langle {uw} \right\rangle } + \frac{{\partial U}}{{\partial z}}{\left\langle {vw} \right\rangle }} \right)}$$

(14f)

$${\left\langle {u\theta } \right\rangle } = - \frac{\tau }{{\alpha _{5} }}{\left( {\frac{{\partial \Uptheta }}{{\partial z}}{\left\langle {uw} \right\rangle } + \alpha _{4} \frac{{\partial U}}{{\partial z}}{\left\langle {w\theta } \right\rangle }} \right)}$$

(14g)

$${\left\langle {v\theta } \right\rangle } = - \frac{\tau }{{\alpha _{5} }}{\left( {\frac{{\partial \Uptheta }}{{\partial z}}{\left\langle {vw} \right\rangle } + \alpha _{4} \frac{{\partial V}}{{\partial z}}{\left\langle {w\theta } \right\rangle }} \right)}$$

(14h)

$${\left\langle {w\theta } \right\rangle } = - \frac{\tau }{{\alpha _{5} }}{\left( {\frac{{\partial \Uptheta }}{{\partial z}}{\left\langle {w^{2} } \right\rangle } - \alpha _{4} \beta g{\left\langle {\theta ^{2} } \right\rangle }} \right)}$$

(14i)

Equations 14a–14i were solved via symbol algebra. Below, we present expressions for those turbulent momentum and heat fluxes that were used in a numerical test to solve the system of Eqs. 13a–13e:

$${\left( {{\left\langle {uw} \right\rangle },\;{\left\langle {vw} \right\rangle }} \right)} = - K_{M} {\left( {\frac{{\partial U}}{{\partial z}},\;\frac{{\partial V}}{{\partial z}}} \right)}$$

(15a)

$${\left\langle {w\theta } \right\rangle } = - K_{H} \frac{{\partial \Uptheta }}{{\partial z}} + \gamma _{c} $$

(15b)

$$K_{M} = E\tau S_{M} \quad K_{H} = E\tau S_{H} $$

(15c)

$$S_{M} = \frac{1}{D}{\left\{ {s_{0} {\left[ {1 + s_{1} G_{H} {\left( {s_{2} - s_{3} G_{H} } \right)}} \right]} + s_{4} s_{5} {\left( {1 + s_{6} G_{H} } \right)}{\left( {\tau \beta g} \right)}^{2} \frac{{{\left\langle {\theta ^{2} } \right\rangle }}}{E}} \right\}}$$

(15d)

$$S_{H} = \frac{1}{D}{\left\{ {\frac{2}{3}\frac{1}{{c_{{1\theta }} }}{\left( {1 + s_{6} G_{H} } \right)}} \right\}}$$

(15e)

where:

$$\gamma _{c} = \frac{1}{D}{\left\{ {1 + \frac{2}{3}\alpha ^{2}_{2} G_{M} + s_{6} G_{H} } \right\}}\alpha _{5} {\left( {\tau \beta g} \right)}{\left\langle {\theta ^{2} } \right\rangle }$$

(15f)

is the countergradient term, which is absent in models with closure levels 2.0 and 2.5 (Mellor and Yamada 1974, 1982; Cheng et al. 2002).

The quantities G _H and G _M are defined as:

$$\begin{aligned} & G_{H} \equiv {\left( {\tau N} \right)}^{2} \quad G_{M} \equiv {\left( {\tau S} \right)}^{2} \\ & N^{2} = \beta g\frac{{\partial \Uptheta }}{{\partial z}}\quad S^{2} \equiv {\left( {\frac{{\partial U}}{{\partial z}}} \right)}^{2} + {\left( {\frac{{\partial V}}{{\partial z}}} \right)}^{2} \\ \end{aligned} $$

(15g)

and for Eqs. 15a–15f, we have:

$$D = 1 + d_{1} G_{M} + d_{2} G_{H} + d_{3} G_{M} G_{H} + d_{4} G^{2}_{H} + {\left[ {d_{5} G^{2}_{H} - d_{6} G_{M} G_{H} } \right]}G_{H} $$

(15h)

$$\begin{aligned} & d_{1} = \frac{2}{3}\alpha ^{2}_{2} ,\;d_{2} = \frac{{10}}{3}\frac{{\alpha _{3} }}{{c_{{1\theta }} }},\;d_{3} = \frac{2}{3}\alpha _{2} \frac{{\alpha _{3} }}{{c_{{1\theta }} }}{\left( {\alpha _{2} - \alpha _{5} } \right)}, \\ & d_{4} = \frac{{11}}{3}{\left( {\frac{{\alpha _{3} }}{{c_{{1\theta }} }}} \right)}^{2} ,\;d_{5} = \frac{4}{3}{\left( {\frac{{\alpha _{3} }}{{c_{{1\theta }} }}} \right)}^{2} ,\;d_{6} = \frac{2}{3}\alpha _{2} \alpha _{5} {\left( {\frac{{\alpha _{3} }}{{c_{{1\theta }} }}} \right)}^{2} , \\ & s_{0} = \frac{2}{3}\alpha _{2} ,\;s_{1} = \frac{1}{{\alpha _{2} }}{\left( {\frac{{\alpha _{3} }}{{c_{{1\theta }} }}} \right)},\;s_{2} = \alpha _{2} - \alpha _{5} ,\;s_{3} = \alpha _{5} {\left( {\frac{{\alpha _{3} }}{{c_{{1\theta }} }}} \right)}, \\ & s_{4} = \alpha _{3} \alpha _{5} ,\;s_{5} = \alpha _{5} + \frac{4}{3}\alpha _{2} ,\;s_{6} = \frac{{\alpha _{3} }}{{c_{{1\theta }} }},\;\alpha _{5} = \frac{{1 - C_{{2\theta }} }}{{C_{{1\theta }} }} \\ \end{aligned} $$

(15i)

The variance of the vertical turbulent velocity and the horizontal heat fluxes are determined from the following expressions:

$${\left\langle {w^{2} } \right\rangle } = \frac{1}{D}{\left\{ {\frac{2}{3}E{\left( {1 + s_{6} G_{H} } \right)} + \frac{4}{3}\alpha _{3} \alpha _{5} {\left( {\tau \beta g} \right)}^{2} {\left\langle \theta \right\rangle }^{2} {\left( {1 - \frac{1}{2}\alpha _{2} \alpha _{5} G_{M} + s_{6} G_{H} } \right)}} \right\}}$$

(15j)

$$\begin{aligned} & {\left\langle {u\theta } \right\rangle } = \frac{1}{D}{\left\{ {\frac{2}{3}E\tau \frac{1}{{c_{{1\theta }} }}{\left[ {\alpha _{2} + {\left( {\alpha _{2} + \alpha _{5} } \right)}s_{6} G_{H} + \alpha _{5} } \right]}\tau \frac{{\partial U}}{{\partial z}}} \right\}}\frac{{\partial \Uptheta }}{{\partial z}} \\ & \quad \quad \quad - \frac{1}{D}\tau \frac{{\partial U}}{{\partial z}}\alpha _{5} {\left( {\tau \beta g} \right)}{\left\langle \theta \right\rangle }^{2} \left\{ {\alpha _{5} {\left( {1 + \frac{2}{3}\alpha ^{2}_{2} G_{M} } \right)} + {\left( {\alpha _{5} - \frac{4}{3}\alpha _{2} } \right)}s_{6} G_{H} } \right. \\ & \quad \quad \quad \left. { + \frac{2}{3}s_{6} \alpha ^{2}_{2} \alpha _{3} G_{M} G_{H} - \frac{4}{3}s^{2}_{6} \alpha _{2} G^{2}_{H} } \right\} \\ \end{aligned} $$

(15k)

$$\begin{aligned} & {\left\langle {v\theta } \right\rangle } = \frac{1}{D}{\left\{ {\frac{2}{3}E\tau \frac{1}{{c_{{1\theta }} }}{\left[ {\alpha _{2} + {\left( {\alpha _{2} + \alpha _{5} } \right)}s_{6} G_{H} + \alpha _{5} } \right]}\tau \frac{{\partial V}}{{\partial z}}} \right\}}\frac{{\partial \Uptheta }}{{\partial z}} \\ & \quad \quad \quad - \frac{1}{D}\tau \frac{{\partial V}}{{\partial z}}\alpha _{5} {\left( {\tau \beta g} \right)}{\left\langle \theta \right\rangle }^{2} \left\{ {\alpha _{5} {\left( {1 + \frac{2}{3}\alpha ^{2}_{2} G_{M} } \right)} + {\left( {\alpha _{5} - \frac{4}{3}\alpha _{2} } \right)}s_{6} G_{H} } \right. \\ & \quad \quad \quad \left. { + \frac{2}{3}s_{6} \alpha ^{2}_{2} \alpha _{3} G_{M} G_{H} - \frac{4}{3}s^{2}_{6} \alpha _{2} G^{2}_{H} } \right\} \\ \end{aligned} $$

(15l)

1.3 Constants of the three-parameter model of turbulence

For the correlations with pressure fluctuations of the dynamic turbulent field, the standard models from Launder et al. (1975) and Launder (1996) are used. These models were successfully applied in solving different problems; therefore, the values of the numerical coefficients in the model expressions for these correlations have been approved sufficiently well. These coefficients are presented in Launder (1996) as the graphical dependence:

$${{\left( {1 - c_{2} } \right)}} \mathord{\left/ {\vphantom {{{\left( {1 - c_{2} } \right)}} {c_{1} }}} \right. \kern-\nulldelimiterspace} {c_{1} } = 0.23$$

(15m)

For the relaxation coefficient in the model of the slow part of pressure–strain correlation Π ⁽¹⁾ _ij (Eq. 7b), the value c ₁ = 2.0 is taken from the commonly used range between 1.5 and 2.0. For c ₁ = 2.0, the numerical value of the coefficient c ₂ found from Eq. 15m is 0.54. When selecting the value of the coefficient c ₃ in the buoyancy terms α ₃ B _ij in Eq. 7b, one can use the solution of simple problems with allowance for buoyancy effects (Gibson and Launder 1978; Lumley and Mansfield 1984) (c ₃ = 0.776). Here, for this coefficient, the value 0.8 is taken, which corresponds to the value obtained by Cheng et al. (2002) via the renormalization group technique. Numerical values of the coefficients in the pressure–temperature correlation Π ^θ _i in Eq. 7c are c _1θ  = 3.28 and c _2θ  = c _3θ  = 0.5. These values are calibrated during the modeling of different turbulent stratified flows, both homogeneous and inhomogeneous (Kurbatskii 2001; Sommer and So 1995). We note that the numerical value coefficient c ₁ calculated by Cheng et al. (2002) with use of the renormalization group technique turned out to be 2.5. At the same time, it should be remembered that, for example, for the widely used E − ɛ model of turbulence, this technique yields the values of the constants appearing in the ɛ equation that are noticeably different from the values calibrated with the database of measurements and commonly used in computations. Numerical coefficients in the diffusion terms have the following values: σ _E  = 1.2, σ _ɛ  = 1.2, and σ _θ  = 0.6. In Eq. 5a, for the TKE dissipation rate, we have ψ ₀ = 3.8, ψ ₁ = ψ ₂ = 2.4, and ψ ₃ = 0.3.

Note that the improved three-parameter anisotropic model for the turbulent ABL includes eight base constants (c ₁, c ₂, c ₃, c _1θ , c _2θ  = c _3θ , ψ ₀, ψ ₁ = ψ ₂, ψ ₃) and three Prandtl numbers (σ _E , σ _ɛ , σ _θ ), which are entered into the models of processes of turbulent diffusion of the transfer equations for the functions E, ɛ, and 〈θ ²〉. The calibrated numerical values of these constants remain invariant during the solution of any problems of atmospheric turbulent flows with the calculation of not only the distributions of average hydrothermodynamic fields, but also the distributions of anisotropic turbulent momentum fluxes (normal and tangential turbulent stresses) and components of the vector of turbulent heat flux. The invariance of numerical values for the set of constants is a natural requirement that must be satisfied in any model that is constructed on the basis of the RANS approximation and that claims to obtain results consistent with the data of measurements and observations for a wide class of problems of stratified atmospheric flows.

1.4 Calculation of the effects of urban roughness on a flow in the ABL

In this numerical test, the urban heat island is modeled through specification of the temperature difference between the urbanized surface and its vicinities and the time-dependent boundary condition for the temperature models a 24-h cycle of heating the Earth’s surface by the Sun. For this reason, the mechanical factors are implemented by the parameterization scheme completely, whereas the contribution to the balance of potential temperature is implemented only approximately (heating/cooling of the surfaces of buildings). Below, for reference, expressions for the source terms that describe the effects of the urbanized surface on air flow are presented.

The loss of momentum (frictional force) at the horizontal surfaces (building tops, street-canyon bottoms) is calculated according to the formula (Martilli et al. 2002):

$$\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{F}u^{{{\left( 1 \right)}}}_{J} = - \frac{{k^{2} }}{{{\left[ {\ln {\left( {\frac{{{\Updelta z} \mathord{\left/ {\vphantom {{\Updelta z} 2}} \right. \kern-\nulldelimiterspace} 2}}{{z_{0} }}} \right)}} \right]}^{2} }}f_{m} {\left( {\frac{{{\Updelta z} \mathord{\left/ {\vphantom {{\Updelta z} 2}} \right. \kern-\nulldelimiterspace} 2}}{{z_{0} }},\;Ri} \right)}{\left| {U_{J} } \right|}\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{U}_{J} S^{H}_{J} $$

(16a)

where f _m is the function of constant fluxes of the MOST, Ri is the volumetric Richardson number, S ^H _J is the total area of the horizontal surfaces on layer J, U _J is the horizontal wind speed on layer J, k = 0.41 is the von Karman constant, and z ₀ is the roughness parameter (z ₀ = 0.1 m in calculations). This formula is used for the urban canopy layer 50 m in height divided into five vertical layers (J = 1,…, 5) with step ∆z = 10 m. Although the use of the MOST of constant fluxes (function f _m ) in the urban canopy layer is difficult to substantiate, this theory is employed due to the lack of an alternative.

The momentum exchange at the vertical surfaces of buildings, which is induced by the forces of pressure and viscous friction, is calculated as in Martilli et al. (2002) via the formula:

$$Fu^{{{\left( 2 \right)}}}_{J} = - C_{D} {\left| {U_{J} } \right|}\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{U}_{J} S^{V}_{J} $$

(16b)

where S _J is the total surface area of building walls on layer J and C _D is the form drag coefficient, which is taken to be equal to 0.4 in accordance with measurements in a wind tunnel with surface roughness in the form of cubes (Raupach 1992). The sum of expressions 16a and 16b divided by the volume of air in a computational mesh yields an expression for the source $\ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{{u_{i} }} .$

The turbulent heat fluxes from the street-canyon bottom (horizontal surfaces of buildings) are calculated via a formula similar to Eq. 16a:

$$\ifmmode\expandafter\vec\else\expandafter\vecabove\fi{F}\theta ^{{{\left( 1 \right)}}}_{J} = - \frac{{k^{2} }}{{{\left[ {\ln {\left( {\frac{{{\Updelta z} \mathord{\left/ {\vphantom {{\Updelta z} 2}} \right. \kern-\nulldelimiterspace} 2}}{{z_{0} }}} \right)}} \right]}^{2} }}f_{h} {\left( {\frac{{{\Updelta z} \mathord{\left/ {\vphantom {{\Updelta z} 2}} \right. \kern-\nulldelimiterspace} 2}}{{z_{0} }},\;Ri} \right)}{\left| {U_{J} } \right|}\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }S^{H}_{J} $$

(16c)

where f _h is the function of turbulent heat flux, which is similar to f _m , and $\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }$ is the deviation of potential temperature. In order to obtain realistic results, it is necessary to take into account the heat exchange between the air and the building wall, whose temperature is specified in our formulation of the problem as T _w  = 0.6T _g , where T _g is the given temperature of the ground surface. In this case, the heat flux for a rigid wall is written following Vu et al. (2002) as:

$$F\theta ^{{{\left( 2 \right)}}}_{J} = c_{h} {\left( {T_{w} - T_{g} } \right)}$$

(16d)

Here, c _h  = (6.15 + 4.18|U|) is the heat exchange coefficient, where U is the average horizontal wind speed. As with the momentum, the source term in the temperature balance equation $\ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{\theta }$ is obtained through the division of the sum of expressions 16c and 16d by the volume of air in a computational mesh.

At the rigid horizontal surfaces, one should take into account the generation of TKE by velocity shear and buoyancy through the use of the values of surface fluxes and the relations of the similarity theory. The generation of TKE by horizontal surfaces is determined via the expression:

$$P^{{{\left( 1 \right)}}}_{{EJ}} = {\left[ { - \frac{{{\left( {{Fu^{{{\left( 1 \right)}}}_{J} } \mathord{\left/ {\vphantom {{Fu^{{{\left( 1 \right)}}}_{J} } {S^{H}_{J} }}} \right. \kern-\nulldelimiterspace} {S^{H}_{J} }} \right)}^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{k{\left( {{\Updelta z} \mathord{\left/ {\vphantom {{\Updelta z} 2}} \right. \kern-\nulldelimiterspace} 2} \right)}}} + \frac{g}{{\Uptheta _{0} }}\frac{{F\theta ^{{{\left( 1 \right)}}}_{J} }}{{S^{H}_{J} }}} \right]}S^{H}_{J} \Updelta z$$

(16e)

Expression (16e) should be added to the standard term of TKE generation by the velocity shear and buoyancy P _E multiplied by the volume of air in a computational mesh:

$${\left( {P_{E} } \right)}_{J} = {\left( {V_{J} - S^{H}_{J} \Updelta z} \right)}P_{E} + P^{{{\left( 1 \right)}}}_{{EJ}} $$

(16f)

The transport equation for the TKE, E, is used:

$$ \frac{\partial E}{{\partial {\kern 1pt} t}} + U\frac{{\partial {\kern 1pt} E}}{{\partial {\kern 1pt} x}} + W\frac{{\partial {\kern 1pt} E}}{{\partial {\kern 1pt} z}} = - \frac{{\partial {\kern 1pt} \langle E^{'} w\rangle }}{{\partial {\kern 1pt} z}} + K_{M} \left[ {\left( {\frac{{\partial {\kern 1pt} U}}{{\partial {\kern 1pt} z}}} \right)^{2} + \left( {\frac{{\partial {\kern 1pt} V}}{{\partial {\kern 1pt} z}}} \right)^{2} + \beta g\langle w\theta \rangle } \right] - \varepsilon + \hat{D}_{E} , $$

(16g)

where:

$$P_{E} = K_{M} {\left[ {{\left( {\frac{{\partial U}}{{\partial z}}} \right)}^{2} + {\left( {\frac{{\partial V}}{{\partial z}}} \right)}^{2} + \beta g{\left\langle {w\theta } \right\rangle }} \right]}$$

is the sum of the shear and buoyant production terms of the TKE. The details of calculating the quantities V _J , S ^V _J , and S ^H _J can be found in Martilli et al. (2002).

The source term $\ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{E}$ in the TKE balance equation is written as the ratio of (F _E ) _J to V _J , and (F _E ) _J is determined on the basis of the arguments used to write a similar expression (13b) for the momentum and has the form:

$${\left( {F_{E} } \right)}_{J} = C_{D} {\left| U \right|}^{3} S^{V}_{J} $$

(16h)

Parameterization (Eq. 16e) is based on the concept that the presence of buildings increases the transformation of the average kinetic energy into the TKE (Raupach and Shaw 1982).

The source term of additional dissipation $\ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{\varepsilon }$ is introduced from evident simple considerations based on the fact that, while increasing the transformation of the average kinetic energy into the TKE, the resulting effect of buildings in the urban canopy layer must also lead simultaneously to an increase in the dissipation of TKE. According to Hiraoka et al. (1989), this additional dissipation is calculated as:

$$ \ifmmode\expandafter\hat\else\expandafter\^\fi{D}_{\varepsilon } = C_{{p\varepsilon }} \varepsilon \frac{{E^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} {{\ell }} $$

(16i)

with the coefficient C _pε  = 0.7 and the linear scale ℓ being calculated according to the formula:

$$ \frac{1} {{\ell }} = {\sum\limits_J {\gamma {\left( {z_{J} } \right)}\frac{1} {{z_{j} }}} } $$

(16j)

where γ(z _J ) is the probability density for buildings of a given height on layer z _J (a specified quantity).

The prognostic equation for 〈θ ²〉 in the models with high-order moments accurately predicts a maximum of this characteristic in the inversion layer near the top of the ABL (see André et al. 1978; Fig. 10). In models of 2.5 level closure, the 〈θ ²〉 transport equation is replaced by the algebraic down-gradient approximation (Cheng et al. 2002). Because of the insufficiency of the information on the influence of a thermal regime in the urban matrix of buildings on the behavior of the temperature variance, the 〈θ ²〉 transport equation is used in this study without the extra term. The present nonlocal model for the vertical turbulent heat flux includes a so called countergradient term γ _C (see Eq. 15b). Holtslag and Boville (1993) showed that the nonlocal transport expressions for the temperature and scalar fluxes are capable of transporting moisture more efficiently from the surface to higher vertical levels in a global climate model. For this reason, the transport equation for a temperature variance is included in the present model of an urban boundary layer, despite it lacking in the extra term.

1.5 Boundary conditions for a system of equations of the ABL (Eqs. 13a–13e)

Following Spanton and Williams (1988), one can find an expression for the ratio of the horizontal wind speeds at the first two computational nodes above the surface (z ₂ > z ₁ > 0). This expression represents the finite difference boundary condition for the wind speed at the lower boundary and is written as:

$$ \frac{{U_{1} }} {{U_{2} }} = \frac{{V_{1} }} {{V_{2} }} = \frac{{\ln {\left( {{z_{1} } \mathord{\left/ {\vphantom {{z_{1} } {z_{0} }}} \right. \kern-\nulldelimiterspace} {z_{0} }} \right)}}} {{\ln {\left( {{z_{2} } \mathord{\left/ {\vphantom {{z_{2} } {z_{0} }}} \right. \kern-\nulldelimiterspace} {z_{0} }} \right)}}} $$

(17a)

In our numerical test, the temperature measured at the ground surface is interpolated with dependence (Eq. 11). Therefore, as in Yamada and Mellor (1975), the boundary condition in a finite difference form is also used for the temperature:

$$ \ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{2} = \Updelta _{\theta } \ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{3} + \ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{g} {\left( {1 - \Updelta _{\theta } } \right)}\quad \Updelta _{\theta } = \frac{{\ln {\left( {{z_{2} } \mathord{\left/ {\vphantom {{z_{2} } {z_{{0t}} }}} \right. \kern-\nulldelimiterspace} {z_{{0t}} }} \right)}}} {{\ln {\left( {{z_{3} } \mathord{\left/ {\vphantom {{z_{3} } {z_{{0t}} }}} \right. \kern-\nulldelimiterspace} {z_{{0t}} }} \right)}}} $$

(17b)

where z _0t  = 0.6z ₀ (z ₀ = 0.1 m in calculations).

In order to calculate turbulent fluxes near the surface, the MOST is used to relate vertical gradients in the surface layer and the empirical function Φ, obtained from the data of the Kansas experiment, is taken from Hiraoka et al. (1989). The boundary conditions are specified at the first (from the surface) computational node and have the form:

$$ E_{1} = {u^{2}_{{ * 0}} } \mathord{\left/ {\vphantom {{u^{2}_{{ * 0}} } {{\sqrt {c_{\mu } } }}}} \right. \kern-\nulldelimiterspace} {{\sqrt {c_{\mu } } }} $$

(17c)

$$ \varepsilon _{1} = u^{3}_{{ * 0}} {\left( {\Upphi \mathord{\left/ {\vphantom {\Upphi {0.41z_{z} }}} \right. \kern-\nulldelimiterspace} {0.41z_{z} } - 1 \mathord{\left/ {\vphantom {1 {0.41L}}} \right. \kern-\nulldelimiterspace} {0.41L}} \right)} $$

(17d)

$$ {\left\langle {\theta ^{2} } \right\rangle } = \theta ^{2}_{{ * 0}} \left\{ {\begin{array}{*{20}l} {{4{\left( {1 - 8.3\varsigma } \right)}^{{ - 2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} } \hfill} & {{\varsigma \leq 0} \hfill} \\ {4 \hfill} & {{\varsigma > 0} \hfill} \\ \end{array} } \right. $$

(17e)

$$ \theta _{{ * 0}} = \frac{k} {{\Pr _{t} }}{\left[ {\ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }{\left( {z_{1} } \right)} - \ifmmode\expandafter\hat\else\expandafter\^\fi{\Uptheta }_{g} } \right]}{\left[ {\ln {{\left( {z_{1} + z_{{0t}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {z_{1} + z_{{0t}} } \right)}} {z_{0} }}} \right. \kern-\nulldelimiterspace} {z_{0} } + \ln {\left( {{z_{0} } \mathord{\left/ {\vphantom {{z_{0} } {z_{{0t}} }}} \right. \kern-\nulldelimiterspace} {z_{{0t}} }} \right)} - \Upphi } \right]}^{{ - 1}} $$

(17f)

$$ \begin{aligned} & \Upphi = \left\{ {\begin{array}{*{20}l} {{{\left( {1 - 15\varsigma } \right)}^{{ - 1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} } \hfill} & {{\varsigma \leq 0} \hfill} \\ {{1 + 5\varsigma } \hfill} & {{\varsigma > 0} \hfill} \\ \end{array} } \right. \\ & \varsigma = z \mathord{\left/ {\vphantom {z L}} \right. \kern-\nulldelimiterspace} L \\ & L = {u^{2}_{{ * 0}} } \mathord{\left/ {\vphantom {{u^{2}_{{ * 0}} } {0.41\beta g\theta _{{ * 0}} }}} \right. \kern-\nulldelimiterspace} {0.41\beta g\theta _{{ * 0}} } \\ \end{aligned} $$

(17g)

The frictional velocity u _*0 and frictional temperature θ _*0 are computed using the MOST and the noniterative scheme of Louis (1979).