A k −
TURBULENCE CLOSURE MODEL FOR THE ATMOSPHERIC
BOUNDARY LAYER INCLUDING URBAN CANOPY
THANH CA VU1
, YASUNOBU ASHIE3 and TAKASHI ASAEDA2
1Department of Civil and Environmental Engineering, and 2Graduate School of Science and
Engineering, Saitama University, Urawa, Saitama 338-8570, Japan; 3Building Research Institute,
Ministry of Construction, 1 Tatehara, Tsukuba, Japan
(Received in final form 28 June 2001)
Abstract. A numerical model for the computation of the wind field, air temperature and humidity in
the atmospheric boundary layer (ABL) including the urban canopy was developed for urban climate
simulation. The governing equations of the model are derived by applying ensemble and spatial
averages to the Navier–Stokes equation, continuity equation and equations for heat and water vapour
transfer in the air. With the spatial averaging procedure, effects of buildings and other urban structures
in the urban canopy can be accounted for by introducing an effective volume function, defined as the
ratio between the volume of air in a computational mesh over the total volume of the mesh. The
improved k−  model accounts for the anisotropy of the turbulence field under density stratification.
In the improved k −  model, the transport of momentum and heat in the vertical direction under
density stratification is evaluated based on the assumption of a near-equilibrium shear flow where
transport effects on the stresses and heat fluxes are negligible. The heating processes at surfaces
of buildings and ground are also modelled. The comparison of the computational results obtained
with the present model and existing observational data and numerical models shows that the present
model is capable of predicting the structure of turbulence in the urban canopy layer under density
stratification. Numerical experiments with the new model show that the flow behaviour of the air in
the urban canopy layer is strongly affected by the existence of buildings and density stratification.
Keywords: Buoyancy effect, Effective volume function, Improved k− model, Spatial and ensemble
averages, Urban canopy, Urban climate.
1. Introduction
It is well-known that the ground surface in the urban area, with a large portion
covered by impermeable pavements such as asphalt and/or concrete, can act as a
heat reservoir to absorb incoming solar radiation during the day. The absorbed solar
radiation together with anthropogenic heat, released by human activities, create the
so-called urban heat island (Oke, 1982, 1988; Oke and Cleugh, 1987; Stull, 1988;
Asaeda and Vu, 1993).
The study of the urban heat island has been divided into two scales. In the micro-
scale approach, the heating characteristics of urban elements such as pavement and
street canyons inside the layer between the roof tops and the ground, known as the

Now at INA Cooperation, 1-44-10, Sekiguchi, Bunkyo-ku, Tokyo 112-8668, Japan.
Boundary-Layer Meteorology 102: 459–490, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

460 THANH CA VU ET AL.
urban canopy layer (Oke, 1982, 1988; Stull, 1988), are investigated (Asaeda and
Vu, 1993; Swaid, 1993; Vu et al., 1994). In the mesoscale approach, the feedback
of individual heat island causual elements to the urban climate is investigated by a
general above-city circulation model (Vukovich et al., 1976; Kimura and Arakawa,
1983; Kondo, 1990).
For a general above-city circulation model, a proper modelling of the lower
portion of the atmospheric boundary layer (ABL) above an urban area plays a
crucial role in providing an accurate solution for the governing equations. In gen-
eral, the surface layer above an urban area can be divided into several sublayers.
The upper sublayer, known as the inertial sublayer (Tennekes, 1973), is the region
where height above the ground is the only length scale in adiabatic conditions, and
where the flow can be described one-dimensionally using surface-layer similarity
theory (Raupach and Thom, 1981). The lower sublayer, as the region close to and
within the canopy layer itself, where the flow is three-dimensional because it is
mechanically and thermally influenced by nearby canopy elements, is called the
roughness sublayer (Raupach and Thom, 1981). Hosker (1985) provided a detailed
review of works concerning the flow and dispersion of pollutant around isolated
structures and building clusters. Among his main conclusions are that the flow field
around a very large building surrounded by small structures will be similar to the
flow expected if the large building has no neighbours, and the smaller structures
seems to act mainly as a kind of enhanced surface roughness; and that, the most
complicated flow problem is that of a true building cluster – a group of buildings
of roughly comparable size. For the latter situation, once a clear physical picture
of the flow and turbulence characteristics is obtained for a building cluster config-
uration, a mathematical model can be developed with some confidence. As cited
by Raupach and Thom (1981), and Raupach and Shaw (1981), despite its three-
dimensional nature, the roughness sublayer can be described using a single vertical
axis if flow and canopy properties are horizontally averaged. However, even with
this, the characteristics of turbulence, air flow and various other physical properties
of the urban canopy must be understood for the proper modelling of the exchange
of momentum, heat and mass between the canopy and outside air. This is extremely
important since sources of pollutants are usually situated inside street canyons
and/or above roofs. For the urban canopy, the reduction of the sky view-factor
inside street canyons can increase the possibility of interception of a large portion
of the longwave radiation, emitted from wall and ground surfaces. The intercepted
longwave radiation together with multiple reflection of radiation and anthropogenic
heat produced by human activities, can make the wind field and heating processes
very complicated. A model for the wind field and heating processes in this region
must account for all of these processes.
From the early 1970s, a large number of turbulent closure models have been de-
veloped for the simulation of physical processes in the atmospheric boundary layer.
Examples of such models are Vukovich et al. (1976), CSUMM (Colorado State
University Mesoscale Model) of Pielke (1974), RAMS (Regional Atmosphere