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ABSTRACT A local-scale one-dimensional model for the prediction of the thermal structure of the Planetary Boundary Layer (PBL) is presented. Time evolution of the surface-layer and whole boundary-layer thickness is considered, together with an

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A MODEL FOR THE EVOLUTION IN RURAL AREAS OF THE TEMPERATURE PROFILE IN THE BOUNDARY LAYER AND A COMPARISON WITH MEASURED DATA
R. SALERNO and G. GIANOTTI
AMBIO Ricerche Ambientali, Corso Lodi 106, 20139 Milan, Italy
Received in final form 22 September, 1994)
Abstract
A local-scale one-dimensional model for the prediction of the thermal structure of the Plan- etary Boundary Layer PBL) is presented. Time evolution of the surface-layer and whole boundary- layer thickness is considered, together with an evaluation of the energy balance at the ground. A thin transition layer TTL) between the soil and the surface boundary layer SL) is introduced to provide a better definition of the bottom boundary condition for temperature T :
T t, z).
Numerical tests of the model have been performed and compared with measured temperature profiles and observed surface data for two places in different seasons. The results show that the surface temperature controls the evolution of the whole profile. The bottom boundary condition is very important in the description of T t, z). Finally it is shown that even a simple model can give realistic results when correct values of the characteristic parameters are used.
1 Introduction
Since variations in the vertical temperature profile T .) in the surface layer depend on what happens near the surface, the evaluation of the energy balance at the ground is fundamental to reproduce the entire profile correctly. The ground characteristics influence the exchange of heat and moisture between the surface and the atmosphere and this modifies the atmospheric profile. Thus the description of T t~ z) essentially requires evaluation of the radiation budget, modelling of the Planetary Boundary Layer PBL) and determination of boundary conditions for the considered area. The behaviour of the boundary layer depends on the surface fluxes, particularly of heat, without which it is not possible to describe PBL dynamics. Some models consider processes between the soil and the atmosphere, using some meteorological variable at a prescribed height Buchan, 1982; Stathers
et al.
1988). Analytical and numerical approaches are also used. The analytical models Lettau, 1951; Kuo, 1968) make certain simplifying assumptions to describe the thermal and moisture behaviour of the atmosphere and the soil. The numerical ones Sasamori, 1968; Estoque, 1973) are generally complete and realistic but require powerful computing facilities. An important parameter for describing surface behaviour is its temperature, Ts. Two approaches are often used for modelling Ts: the force- restore and the partial differential equation methods Johnson
et al.
1991). In recent years different authors Acs
et al.
1991; Yoshida and Kunitomo, 1986) have attempted to model the evolution of the temperature profile in the PBL using a one-dimensional approach. Following this example, the aim of this paper is to develop and test a one-dimensional time-dependent model of temperature in
Boundary-LayerMeteorology
73: 255-278, 1995. ~) 1995
Kluwer Academic Publishers. Printed in the Netherlands.
256
R. SALERNO AND G. GIANOTI I
the PBL. It should be pointed that the development of
T t, z)
requires a correct description of the evolution of the characteristic layers of the PBL. Thus it is important to simulate boundary-layer structure, which requires parameterisation of the PBL itself. Many authors (Deardorff, 1974; Yamada and Mellor, 1975; Mailhot and Benoit, 1982) have attempted this, using data from classical field experiments such as Wangara (Clarke, 1971). An interesting simulation of the evolution of the averaged vertical potential temperature in the boundary layer and a model with an explicit reproduction of the sub-grid scale fluxes has been presented by Andr6
et al.
(1978). In the model presented here, Deardorff s approach for calculating the height of the PBL has been used. Modelling of the surface fluxes, which is the other fundamental characteristic of this model, has been made considering the exchange of heat and moisture between the ground and the atmosphere. The blending of the computation of the PBL heights and the calculation of the surface fluxes, together with other features, has led to quite a simple but realistic model. Its basic structure may be summarized as follows: a) A vertical stratification of the PBL in the Surface Layer (SL) and the Mixed Layer (ML) is adopted, considering the time evolution of the thickness for both the SL and the ML; b) The model is one-dimensional and at a local scale; the atmospheric layers and the ground surface are parallel horizontal planes (Novak, 1991); the soil and the SL are separated by a thin transitional layer (TTL) (Figure 1); c) The main energy processes occur at the ground and solar radiation is an extemal source (Yoshida and Kunitomo, 1986; Johnson
et al.,
1991; Carlson and Boland, 1978); d) The force-restore approach (Bhumralkar, 1975; Blackadar, 1976) has been used; in this, the transfer of thermal energy into mechanical energy and the transfer of thermal energy into phase-transition energy (latent heat) are considered, together with long-wave (ground-air) parameterisation and the absorption of part of the short-wave incident radiation by water vapour; e) The hypothesis of local thermodynamic equilibrium is used; thermal effects due to CO2 and 03 aerosols and other pollutants have not been included; the model does not treat cloud cover situations and the effect of a vegetation layer has not been expressly considered. Model results have been compared with field measurements to examine the model capability to describe the evolution of temperature profiles over a 24 hour period.
2 The model
2.1. PBL EVOLUTION The formulation for wind, temperature and moisture z-dependence in the SL usually takes its lower boundary conditions at a defined height, z0, the roughness length. This height is generally different for wind, temperature and moisture. Regarding
A MODEL FOR THE EVOLUTION IN RURAL AREAS OF THE TEMPERATURE PROFILE
257
top of P L
Mixed Layer
Surface
Layer Thin Transition Layer L 0
SOIL
I I s I
Fig. 1. Coarse representation of the Thin Transition Layer TTL) between the soil and the surface layer, In this layer, the different aspects of thermal exchanges are present.
temperature, models of the thermal balance at the ground always take T~ into account. Hence, there is a sort of discontinuity between the temperature at the ground surface and at the bottom of the SL. In our model, on the other hand, a layer between the soil and the SL has been taken into account. In this Thin Transition Layer TTL), the conductive effect is in some way retained together with the turbulent one to avoid an abrupt step jump from the ground surface to the SL Garratt and Hicks, 1973; Carlson and Boland, 1978). Simpler methods such as interpolation cannot be used because the phenomena are not linear. The depth of this layer should have a characteristic value, which is not necessarily the same as z0. This length has been assumed to be independent of surface characteristics. The bottom boundary condition for SL temperature is now taken at the top of the TTL, where its value is driven by the surface temperature by means of the simple equation derived in Subsection 2.3.
2.1.1. Surface Layer
For the Surface Layer, the following equations are used:
V= ~
In -~ 0=00+-- n -~h -~h
1)
2)
258
R SALERNO AND G GIANOTFI
q=q0+~- In --~h --~h , (3) where the subscript zero refers to the lower boundary; u. represents the friction velocity; 0. and q. are respectively the scale potential temperature and scale specific humidity; z is the vertical coordinate, ~ is a momentum statistical function and 9 a heat statistical function (Businger, 1973). u., 0. and q. are evaluated by inverting the average profiles (1)-(3); L is the Monin-Obukhov length (see Equation 26), k is the yon Karman constant and z0 is the roughness length.
2.1.2. Mixed Layer (ML)
Above the SL, the hydrostatic prediction set may be fully solved to obtain a three- dimensional representation of wind, temperature and humidity evolution (Pielke, 1984):
du 10p d--t - fv - - p Ox ' dv 10p d-~ - fu - p Oy
(4)
5)
Op
o~ = -Pg (6)
w Opo
- V-V, 7)
po Oz p = pRT,
(8) dT 1 dp
- + Q. 9)
cp dt p dt
In the above set of equations, f is the Coriolis parameter; P0 is the average horizontal density; cp is the specific heat at constant pressure p; V is the wind vector and u and v its horizontal components; w is the vertical speed; Q takes into account the heat exchanges. The system of Equations (4)-(9), with initial and boundary conditions (Pielke, 1984), is well known and will not be discussed further. In our case, a main task is to solve Equation (9), which represents the first law of thermodynamics for the system. For the one-dimensional case in the PBL, it can be written as
oo o z oo 1 0(2 at - (o w ) - wN + ~p ~ (lO)
where 0 is the potential temperature and the prime superscript indicates the fluctu- ations. Q and Q are linked by the following expression:
(~
_
OQ
(lOa)
p Oz
A MODEL FOR THE EVOLUTION IN RURAL AREAS OF THE TEMPERATURE PROFILE
259
Data input and initialization
I
_t Computation of humidity and
temperature at ground Eqs. 16-17) I Temperature in the TTL 23) q.
I
Computation of wind and temperature in the TL Eqs. lO-12-29a-29b} Computation of similarity parameters Eqs. 24-25-26) I Computation of characteristic heights Eqs. 13-14-15) J~Computation of total irradiance Eq. 27)
Fig. 2. Model flow chart with equations used for each block.
K-Theory gives a first-order relationship between vertical covariance and the gradients of the averaged variables: 00 (0 w ) =
-K~. 11)
Substituting Equation (11) in (10) and assuming the relationship Kh = K,~ between turbulent transport coefficients, it becomes:
oo 02o oo 0K 1 0Q
Ot -- OZ 2 4- OZ \ OZ Wj 4- ---- pep 9Z
12)
In a one-dimensional model, it is difficult to calculate the vertical speed w. Thus, taking into account the scale analysis of the terms in the Equation (12), a value for [wl = 0.01 m/s has been assumed. Equation (12) needs initial and boundary conditions. A semi-implicit method for the numerical integration of partial differential equations has been used. The parabolic partial differential equation describing turbulent diffusion in the mixed layer is handled with the Crank-Nicholson method, using a 10 min time step and a 25 m vertical step. In Figure 2, the flow chart of the model is shown.
2.1.3. Characteristic heights
Boundary conditions for Equation (12) have to be applied on a changeable boundary linked to the variability of characteristic heights. The following expression (Sharma
et al.
1976) provides the determination of the SL height h:
h = o.2 13)
J i#)

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