a two-dimensional mesoscale numerical model of an urban mixed layerâi. model formulation, surface...

Atmospherw Environment Vol. 24A, No. 4, pp. 829 844, 1990, 0004 6981/90 S3 00+0.00 Printed in Great Britain. i 1990 Pergamon Press plc

A TWO-DIMENSIONAL MESOSCALE NUMERICAL MODEL OF AN URBAN MIXED LAYER--I. MODEL FORMULATION,

SURFACE ENERGY BUDGET, AND MIXED LAYER DYNAMICS

D. W. BYUN* and S. P. S. ARYA Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Raleigh,

NC 27695-8208, U.S.A.

(First received 1 February 1988 and in final form 7 Auyust 1989)

Abstract A two-dimensional, numerical urban mixed layer model, which takes into account differences between the urban and the rural surfaces, as well as roughness and topographical variations over the model domain, is developed. The various land use types over the metropolitan area are specified by different proportions of concrete and grassy soil surfaces. Thermal characteristics of the vegetated soil surface are parameterized in terms of the surface soil moisture content. Distinct differences in the diurnal patterns of urban and rural surface energy budgets, which are found in some observational studies, are also obtained in some of our model case studies. Dynamics of the dome-shaped urban mixed layer is studied for the case of the stationary balanced wind. The balanced wind flow simulation can explain some of the important flow perturbations due to the urban heat island and increased roughness of the urban area.

Key word index: Mesoscale modeling, mixed layer, urban boundary layer, urban heat island, surface energy budget, mixing height.

1, INTRODUCTION

Air quality models are used to simulate the transport and diffusion of air pollutants emitted in the atmosphere. Atmospheric dispersion depends on the turbulence structure of the planetary boundary layer (PBL) in which pollutants are released from near surface sources. Mixing height and mixed-layer transport winds are most important meteorological parameters determining the ground level concentrations. To have an accurate assessment of urban air quality, we need a model which can provide temporal and spatial distributions of mixing height and mean transport winds in the urban boundary layer (UBL) and, at the same time, which is simple enough to be used in conjunction with an urban air quality model. Other applications of UBL modeling are in urban planning. Because of many observational evidences of local climate changes due to urbanization, urban planners have become interested in the cause-and-effect relationships between urbanization, and local climate modification. Urban atmospheric models can be used to study such relationships and also simulate spatial variations of temperature (urban heat island), mean flow, surface fluxes and turbulence characteristics.

Spatial inhomogeneity of the urban area and its surroundings (suburban and rural) precludes utiliz- ation of horizontally-homogeneous PBL models to predict the structure of UBL. In this paper, a simple, two dimensional, mixed-layer model of the time-

*Present affiliation: Computer Sciences Corporation, Re- search Triangle Park, NC 27713, U.S.A.

dependent, horizontally nonhomogeneous UBL, in- cluding different land uses and terrain features, is developed. It is intended for simulating mixed layer winds, atmospheric stability and mixing heights for use in urban air quality models. The mixed layer assumption in the model precludes simulation of vertical wind shear and temperature gradients in UBL and limits the evolution of the urban mixed layer during the daytime period, under fair weather conditions. It is similar to the models of Keyser and Anthes (1977) and Deardorff et al. (1984), but without the overlying free-atmospheric layers above the mixed layer. Different parameterizations of the entrainment process, stability effects and terrain-dependent transfer coefficients and the adoption of the force-restore- method for computing surface temperature in our model are the main distinctions of the present model from previous models. Some investigators (Anthes et al., 1980, 1982) have suggested, on the basis of their larger mesoscale (1000 km) model studies, that mixed layer models are likely to err in their predictions of mixed layer flow, because the latter do not simulate the upper layer dynamics and feedbacks of the same on the boundary layer. However, the horizontal length scale of the UBL is usually much smaller than the scales that are likely to cause significant feedback between boundary layer and upper atmosphere. Therefore, effects of the upper layer perturbation are ignored here. In order to simulate the UBL, a careful pre- scription of physical properties of surface media is required. The actual surface is composed of varying combinations of different surface media, but we sim- plify this in our model by characterizing the surface in

829

830 D.W. BYUN and S. P. S. ARYA

terms of a few representative materials. Our UBL model is primarily driven by large-scale geostrophic winds, but it accounts for different land use types, surface heating and gentle topographical changes. Model formulation, differences in the urban and rural surface energy budget and mixed layer dynamics are presented here. The results of a particular simulation of the St. Louis urban mixed layer will be presented in a companion paper, Byun and Arya (1990).

2. MODEL FORMULATION FOR THE HORIZONTALLY-

INHOMOGENEOUS MIXED LAYER

The mixed layer assumption is approximately valid for the atmosphere between the top of the surface layer (6s) and the base of the elevated capping inversion (r/), especially during day time hours under clear sky when convective activity prevails. Although, an urban atmospheric surface layer is very difficult to define in the presence of tall and nonuniform buildings and other roughness elements, we use the concept to distinguish the part of the boundary layer immediately above the surface which shows rather rapid change of wind speed and temperature from the upper part of the UBL with much smaller variations in wind and potential temperature. Since the surface layer is not specially resolved in the model, the model cannot simulate variations of mean meteorological variables or fluxes in this layer. For practical purposes, we assume that the surface layer occupies no more than 10% of the urban boundary layer. We also assume stronger wind and temperature variations, that are associated with the interfacial inversion layer at the top of the mixed layer, are concentrated in a very thin layer, allowing for a mathematical representation of zeroth-order discon- tinuity across the interface. Radiative flux divergences in the mixed layer and across the interface are ignored here. A schematic of the simplified mixed layer is given in Fig. 1.

The free atmospheric variables at the top of the mixed layer are prescribed in the model. Following Gutman and Berkofsky (1985), we specify the distribution of potential temperature in the free atmosphere as

O=O0+OH(x , y)+~z (1)

where ®0 is a reference surface potential temperature, On(x, y) is the horizontally varying component, and 7 = dO/dz is the potential temperature gradient in the free atmosphere. Although, V may vary slightly over an urban area, for the purpose of simulating the urban mixed layer, horizontal variations of 7 are neglected. For the case in which 8®/8z varies considerably with height, we are mainly interested in the layer just above the mixed layer. For convenience, we express ®o + On(X, y)= ®oo where ®oo is related to ® + (temperature at the inversion base r/(x, y), just above the mixed layer) as

O+ = Ooo(X, y)+ 7tl(x, y). (2)

Thus, the accuracy of upper boundary condition for potential temperature depends on how well one can prescribe ®oo field. Wind at just above the mixed layer is assumed to be in local geostrophical balance.

V + = VGL (/7) + VTO 0 (/~, X, y) (3)

where VaL is large scale geostrophic wind at height z =r/(x, y) and VToo represents the local thermal wind due to spatial changes in Ooo. For a mixed layer under a steady synoptic high, the large scale pressure distribution is close to barotropic, and the dependence of VaL on the mixed layer height can be dropped.

Integrating the usual conservation equations of motion, heat and moisture across the mixed layer (z from 6s to r/) one obtains the governing set of equations for the mean mixed layer wind components (Urn, Vm), potential temperature (®m), specific humidity (Qm) and mixed layer height (h):

D H U m ,6~ + ( gh ~ O m

Dt = - # ~ x x \ 2 ~ o ] ~x

1 .-I-f(V m - Vam)'-I--~[Zx(rl)--zx(6s)'] (4)

D. Vm ,~,7 [ # "~Om

I --f(Um-- Uc,,,)+~-~[zr(tl)--zy(a,)] (5)

Free Atmosphere

Mixed Layer / / ' / h

i S s S

s S

e~

v . -.,,aF,----

o.

Fig. 1. Schematic of the nonhomogeneous urban boundary layer.

Two-dimensional mesoscale numerical model 831

DH Om

Dt

DHQm

Dt

Dnh

Dt

with

Dn()

1 ph [ n , - us] (6)

1 ph [E, - E~ ] (7)

_ _ _ _ h F ~ U m + O V m ] + k~-~ -~-yA w~+ we (8)

e()+_ _ _ : Um o( ) + Vm °( ) Dt et ~-x ~y

rl=h+6s

z x (q) = - pfi-ff(r/) = pA U We

Zy(r/) = -pb--ff(rl)=pAVWe

H, = pwO(rl) = -- pA® W e

E, = p~-q(rl) = -- pAQ W e

AU = U + - U m

AV= V + - V m

A ® = ® + - - 0 m

AQ=Q+ _Qm

U + = UGL(~ ) -~- UT00(X, y, r/)= U~L

a°°°l fLOoo+yas ay _]

V+ = VGL0'])"~- VT00(/, y, r/)= VOL+gF r/--6, 0 0 o o ] J

O + = Oo0 (x, y) + 7q (x, y)

Q + - constant

g' = g(AO/O + ) ~ g(AO/Oo)

(gem, V~m)=; (UGL, V~Odz. s

Wh is the large scale vertical velocity at the top of the mixed layer and W e is the entrainment rate which will be discussed in the following section. To form a closed set of governing equations, the surface fluxes and entrainment rate need to be specified or parameterized in terms of mean variables. Following the parametric relations predicted by Deardorff (1972), Arya (1977, 1984) and others, the surface fluxes of momentum and heat can be expressed in terms of drag and heat transfer coefficients (Ca, Ch) and mean mixed-layer variables as:

Tx (3s) /p ~ 2 2 1/2 = --Uw~=Cd(Um+ V~,) Um (9a)

ry(Ss)/p = - v-ff~ = Cd(U 2 + V2) 1/2 V m (9b)

U s = ~ s = C h ( U 2 - ~ - V2m) 1/2 (Os - - Ore) (10)

Es/p=%-q~=CqM(U~ + V2m)'/Z(Qs-Qm). (11)

For the estimation of Co and Ch, transfer relations based on similarity matching arguments (Arya, 1977;

Deardorff, 1972) are used neglecting baroclinicity and other effects of nonhomogeneity. The transfer coefficients are functions of mixed-layer height, surface roughness, Coriolis parameter, friction velocity and Monin-Obukhov length. For more detailed specifica- tions of Cd and Ch, refer to Byun and Arya (1986). Cq is assumed to have the same value as C h. M is the representative moisture availability of a grid point and Qs is the saturation specific humidity at the surface temperature T s. When the grid-scale wind speed is virtually zero, the minimum wind speed is set to be 0.1 ms - t in Equations (10) and (11). Following Byun and Arya (1986) the entrainment rate is computed with,

3 V h N ] Dnh T°v*LCF-C° J

We= - Wh= (12) Dt ghAO + C T T O V2,

2 2 2 where V2, = W, + ~ U , , N is the Brunt-V~iisfil~i fie- quency for the upper stable atmosphere and CF, CD, CT and rt are constants. In the computation of the convective velocity W,, surface buoyancy flux correc- ted for humidity is used. Following Zeman's (1975)

model results, as well as observed range of -wO~/wO~ in the atmosphere, we have used the value of CF =0.5, Cr=3.55, CD=0.024 and ~/=2.

2.1. Computation of surface temperature

The sensible and latent heat fluxes as well as the anthropogenic heat flux may play important roles in the evolution of the UBL. Since the surface fluxes are parameterized using the bulk transfer method, good estimations of surface temperature and humidity are essential. For the computation of these, we use the so called 'force restore method'. Rate equations for surface temperature (Ts) and ground soil moisture (% volume of liquid water per unit volume of wet soil) are given as (Deardorff, 1978):

0T~ 2x/~-G 2~ (Ts-- Td) (13)

c3t pscd 1 z 1

Ows Es -- p (% - w d) - - = - - c T - - - c 2 (14) Ot pwd' 1 z 1

where dt and d'~ are, respectively, the depths to which diurnal temperature and moisture waves penetrate, ps is the density of surface medium, Pw is the density of liquid water and c is the specific heat of soil per unit mass. The product p~cd 1 is expressed as a function of the soil thermal inertia PT ~ ( J ' s P s C ) 1/2 and the diurnal wave period Za, such that pscda=PTr~/z , where 2 s represents the thermal conductivity of the soil. Also, G is heat flux to or from the soil sublayer, T d is the deep soil temperature, estimated with the mean surface temperature of the previous day. E s is the surface moisture flux, P is the rate of percolation at the surface and w s is the volumetric surface water content. The supplementary equation for the deep soil water con-


tent (Wd) is given as:

t~Wd Es -- P - - ( 1 5 )

at pwd'2 '

d~ is recognized as the depth below which the moisture flux is negligible.

In the present urban mixed layer model, condensa- tion and precipitation processes are not taken into account. Each land use is assumed to consist of specified proportions of surface media, so that we could assign certain values of soil properties to each land use type. Disregarding dependency of the coefficients c t and c 2 in Equation (14) on soil type, values suggested by Deardorff (1978) are used here. Soil heat flux G is estimated from the energy balance equation for a uniform surface:

RN+ RF= Hs + LE + G (16)

where RN is net radiation at the interface, R r represents anthropogenic heat, H s and LE are sensible and latent heat fluxes, respectively. Here, we have neglected any biogenic heat storage due to photosynthetic activity and heat advection in horizontal direction. Net radiation (RN) is estimated from the parameterizations of the net shortwave radiation (Rs), infrared radiations from the earth surface (RL) and the atmosphere (RA), which are described in detail by Byun (1987).

2.2. Land use dependent surface parameters

In the present numerical model, we parameterize each land use type of the St. Louis metropolitan area in terms of a few surrogate surface media. For the modeling purpose, a land use type is assumed to be composed of only two contrasting surface media viz., concrete and grassy soil surface. Doll et al. (1985) presented some results showing similarity between the energy budgets of an urban industrial site with a concrete surface, and a rural site with a vegetated soil surface, using the net radiation data from St. Charles County Airport which is located northwest of the St. Louis metropolitan area. For simplification, different land use types are characterized by different proportions of concrete and grassy soil surfaces. Following Auer's (1978) classification, percentage compositions of the two surface media for each land use type are

Table 1. Different land use types of St. Louis and their percentage compositions of concrete and vegetated soil

Composition Land use type % of concrete % of vegetated

soil

I1, I2 (Industrial) 95 5 C1, R2, R3 (Commercial and compact residential) 80 20 R1 (Common residential) 30 70 R4, AI (Metropolitan natural) 15 85

A2, A3, A4 (Rural) 5 95

summarized in Table 1. Water surface (A5) is treated separately from other land use types. Note that our objective is to facilitate the numerical simulation of the urban boundary layer with simple parameterizations discussed in the earlier section. Surface parameters that dependent on land use type are surface albedo, emissivity, thermal inertia, moisture availability and surface roughness.

The representative values of the various parameters that depend on the surface medium are determined using simple interpolation relations such as:

XLU = v X C + (1 - v) X s (17)

where XLU, Xc and X s represent the values of a surface parameter for the specified land use type, concrete and soil, respectively, and v is the fraction of concrete surface per unit area of that land use type. In the numerical model, the value of the parameter at any grid point is related to its values for soil and concrete as

5 Xg rid= E l/Vi[viXc+-(1-vl)Xs] (18)

i=1

where W~ is the fraction of each land use in a unit square area surrounding the grid point. The fraction of water surface occupying any grid area was assigned appropriate thermal properties of water. Methods of estimating radiation and thermal properties of surface media are briefly discussed below.

Surface albedo and emissivity. Surface albedo depends on the surface medium, as well as on the solar zenith angle. White et al. (1978) reported daytime variation of surface albedo for each land use type for the St. Louis metropolitan area; their results have been incorporated into the present numerical model. Sur- face emissivity for each land use type is estimated using Equation (17) and with the fraction of concrete and vegetation given in Table 1. The prescribed values of emissivity are: for concrete surface, egc=0.82, for grassy soil surface, %s=0.97 and for water surface ew = 0.98. Emissivity of the urban surface decreases as the fraction of the build-up area increases. Consequently, the urban area loses less heat through radiation than the surrounding rural areas.

Soil thermal parameters. Thermal inertia is a meas- ure of how readily heat is transferred through the medium. Temperature variation of the subsurface medium which is subject to heating or cooling is inversely proportional to its thermal inertia. Since the thermal inertia is defined as the square root of product of volumetric soil heat capacity (Cs) and thermal conductivity of the soil (2s), we need to estimate C s and 2 s for each land use type. Soil heat capacity per unit volume is defined as the product of specific heat (c) of a soil and its density (Psw); i.e. Cs=pswC. The specific heat capacity and thermal conductivity are determined from the empirical relations given by McGaw (1979) and Dickinson et al. (1981). If water content by weight (ws) exceeds porosity (Wp), the soil becomes saturated and acts like a water surface for evapor-

Two-dimensional mesoscale numerical model

Table 2. St. Louis land use types and their characteristic roughness length

Land use type Description Roughness (m)

II, I2 Industrial; one five story buildings 1.5 C1, R2, R3 Commercial and compact residential; two ten

story buildings 1.8 RI Common residential; one story buildings 1.0 R4, A1 Metropolitan natural; suburb, parks, cam-

puses 0.5 A2, A3, A4 Rural; heavy vegetation 0.2 A5 Water surface; river, lake 0.005

833

ation. Therefore, moisture availability for soil is given as Ms=min (1, ws/wp). On the other hand, concrete surface is assumed to be air-dry all the time, so that M c --- 0, i.e. there is no moisture transfer from the concrete surface to air and vice versa.

Surface roughness. In the computation of transfer coefficients appearing in the parameterization of surface fluxes (see e.g. Byun and Arya, 1986), the surface roughness parameter must be prescribed at each grid point. Determination of the surface roughness of urban areas with varying land use is very difficult; it depends on the areal distribution, height and density of different types of roughness elements (buildings, trees, grass, concrete, etc.). A very simple method of estimating surface roughness is used in our UBL model. Following Auer's (1978) classification of land use types for the St. Louis metropolitan area and proposed empirical relations between roughness length and land use type in the literature (e.g. Daven- port, 1960: Brutsaert, 1982: Pielke, 1984: Wieringa, 1986), characteristic roughness length (ZoL) was estimated for each land use and given in Table 2. The topography-induced roughness length is estimated from a relation similar to that used by Vukovich et al. (1976).

ZoT =0.001 6(x, y) (19)

where ~(x, y) is the terrain height of the topography above the local flat terrain. The magnitude of multipli- cation coefficient is reduced by one third from Vukov- ich et al.'s (1976) value as land-use related roughness is considered separately here. The overall roughness for the area with topography is obtained from:

z 0 = Zox + Z0L (20)

3. NUMERICAL SCHEMES

The governing set of equations for the horizontally inhomogeneous mixed layer is solved numerically, using the time splitting technique. Here, we closely follow Marchuk's (1974) approach of 'natural ' splitting into three stages, or categories, viz., advection, adjustment (gravity and inertial oscillations) and turbulent exchange (diffusion and friction) stages. These stages refer to only numerical procedures and not to the actual physical processes which are, in fact,

concurrent. Additional spatial filtering (smoothing) is applied for every 30 rain of time integration, crudely simulating the horizontal diffusion processes which were omitted in the derivation of the governing set of equations. The numerical method chosen for each stage is described in the following.

3.1. Adjustment stave

This stage basically deals with geostrophic adjustment of flow under an approximate balance between pressure gradient, Coriolis, and gravitational forces. For the horizontally inhomogeneous mixed layer, it is represented by:

t~Urn ,t~ / /gh x~t~[~ m

f gh ~a(~ m 6~Vm63t : --g't~l'] ' t ~ ) ~ - y --f(Um -- UGm ) = d ' - t T y (21)

Oh [-~Um ~VmG hLV+W/

Notice that the set of Equations (21) is similar to the governing set of equations for a two-layered system with shallow water approximation, which includes the effect of horizontal temperature inhomogeneity as well as pressure gradient forcing. Assuming that the mixed layer height and temperature distributions are given from the previous time step, we use the following finite-difference approximations to Equations (21).

For the Coriolis terms, the numerical scheme is found to be neutrally stable as long as (At)Zf2~<4 (Pielke, 1984) and for the thermally modified pressure gradient and velocity divergence, the scheme is stable for CoAt/Ax ~< 2 where Co is the fastest phase speed of the gravity wave admitted in the system.

un+l n gfl7~+ ,,S - - Ui,j l,j--~i- l,j At 2Ax

{oh" "~®7+1,s-O7 l,s 2Ax

+ftvT,~- Yore)

V~+" " g2lT, s+i -qT , s , ,,s - Vi.J_ At 2Ay

834 D.W. BvuN and S. P. S. ARYA

\ 2 ® 0 / 2Ay

- f (U i".~ ~ - U~m) (22)

hn+ l_h~ [~ ,+1 x ~+1 n U i + l , j - - U i - l , J i,j t , j= __hl, j At

n+l 1 + vi~7.~+ ~ - v , , j + ~

2Ay

3.2. Flux exchange stage

At this stage of computation the effects of surface fluxes and entrainment at the top of the mixed layer are taken into account. The influence of entrainment and large scale subsidence on the mixed layer thick- ness is recognized as the thermodynamic adjustment process rather than the geostrophic adjustment process for the thedmodynamically evolving mixed layer. To integrate the system of first-order ordinary differential equations, the Runge-Kutta formulas of orders 4 and 5 are used.

3.3. Advection stage

For a horizontally inhomogeneous mixed layer, the mean flow advection is an important process to transfer quantities from one place to another. Two- dimensional advection is represented as a series of one-dimensional operations in the time-splitting technique. For convenience, momentum advection is first computed and then scalar advection is calculated. For the one-dimensional advection, an upstream cubic spline scheme, which preserves amplitude and phase extremely well (e.g. Purnell, 1976; Long and Pepper, 1981), is used in the present model. In this scheme, the advectional changes are estimated by the values of the cubic spline at an upstream distance from each grid point. Various types of cubic spline interpolation formulas exist. For example, Pielke (1984) has presented a cubic spline scheme involving first order derivatives of the cubic spline function. It was success- fully adapted for some meso-scale numerical models (e.g. Mahrer and Pielke, 1978). The scheme provides a selective damping of dispersive shortwaves. However, it has been known that the damping by the upwind cubic spline scheme cannot be controlled arbitrarily (e.g. Long and Pepper, 1981). Here, we use a slightly different cubic spline algorithm which can provide more controlled damping, if necessary, to the short waves. This property is beneficial for some numerical studies, such as the UBL simulation where small scale differential spatial heating can produce considerable amount of shortwaves which, in turn, give 'shocks' to the system. Duris (1980) presented a discrete cubic spline interpolation algorithm (ACM ALGORITHM 547) for use in data fitting and smoothing. In his algorithm, the array of coefficients of the second order term in the cubic equation is first solved and then other coefficients are determined. The main difference of this approach from that of Pielke (1984) is that

derivatives in the solution process are discretized with a fixed interpolation step size which can be smaller than the grid distance between nodes. By choosing different step sizes, the amount of smoothing can be controlled; for a larger step size, the smoothing becomes larger and vice versa. Another benefit of this method is that it satisfies second derivative end conditions, as well as first derivative or periodic boundary conditions. For details of this algorithm, refer to Byun (1987) and Duris (1980).

3.4. Spatial filtering

In the present urban mixed layer formulation, transfer of mean variables in horizontal direction is quite limited, because horizontal turbulent fluxes are neglected. This results in considerable differences in the vertical fluxes into the columns of air above quite distinct but adjacent land use types. As time integration goes on, these differences become large and unrealistic horizontal gradients in mean fields are produced. Whereas, in the actual mixed layer, thermal convection has the ability to mix quantities in the horizontal direction as well as in vertical. In this respect, a simple mixed layer formulation is unable to simulate the actual UBL. Instead of going through the parameterization of the neglected horizontal fluxes, we use a spatial filter for the purpose of smearing out local peak values as might happen in actual mixing process.

For the smoothing filter, we use the discrete cubic spline smoothing routine DCSSMO (ACM ALGORITHM 547, Duris, 1980). Smoothing is applied for every 30 min of integration. The amount of smoothing is limited to have a correlation of 0.995 or above between unfiltered and filtered fields.

4. SIMULATION OF THE SURFACE ENERGY BUDGET: URBAN vs RURAL

In a urban boundary layer study, one needs to know how the surface energy is partitioned into various components. It is also necessary to know basic differences in the energy budgets of urban and rural areas. There are a few observational studies comparing urban and rural energy budgets (Cleugh and Oke, 1986; Doll et al., 1985). The energy budget of an infir~itesimally thin soil-atmosphere interface may be assumed to be well described by Equation (16), even in the presence of horizontal advection. Therefore, for simplicity, the present numerical model, without the advection terms, is used to simulate surface energy budgets of extensive urban, suburban and rural surfaces and to explain their sensitivity to several important surface parameters. In particular, sensitivity of the simulated surface energy budget to the soil water content is investigated using the force-restore method for surface temperature, in combination with the mixed layer model.


4.1. Effect of soil moisture

Soil water content is by far the most important surface parameter which influences the thermal behavior of a surface medium. Soil heat conductivity, density and thermal inertia differ significantly for different amounts of soil water content. The value of thermal inertia can differ by a factor of 3 or more, depending on the soil water content. In addition, the ratio of volumetric soil water content to the soil porosity determines the moisture availability at the surface. The result is that the latent heat exchange and, hence, the energy balance of the surface are altered dramatically by similar changes in the soil moisture content. In our UBL model, the soil water content is predicted by Equations (14) and (15). Factors influ- encing the computed soil moisture at any time are the initial values of surface and deep soil water contents (w s and Wd) , rate of percolation (P) at surface, and the rate of evaporation which, in turn, depends on the soil moisture availability and mixed layer dynamics. How- ever, for simulation times of a day or less, the deep soil water content can be regarded as a constant, parti- cularly when there are no precipitation events which might influence it. Therefore, sensitivity of the simulated energy budget to only changes in the initial value of w s is examined here. The deep soil moisture content w o is assumed to be 70% of soil porosity which is set to be 0.5 for the test soil, i.e. wd=0.35.

Figures 2a-c show sensitivities of the maxima in surface energy budget components, mixed layer height and surface temperature to changes in the initial value of the surface soil moisture, for a fixed percolation rate of P- - 0.00005 kg m - 2 s- I. The maximum ground heat flux does not change much with w s. But, the maximum sensible heat flux decreases while the maximum latent heat flux increases as the initial w~ is increased. Consequently, both the final mixed layer height and maximum surface temperature decrease as w s increases. However, the times at which the maximum values occur vary for different parameters. The time of maximum latent heat flux (LE) occurs later for higher initial w~, while the time of maximum ground heat flux (G) occurs earlier. The time of maximum sensible heat flux (H~) increases with the increase in w~ up to 0.2, but it decreases with further increase of w, beyond 0.2. The time of maximum surface temperature behaves similarly to sensible heat flux, but with the initial w~ = 0.3 being the dividing point.

4.2. Simulated energy budget: urban vs rural

The main difference between the energy budgets of urban and rural areas is the way net radiation is partitioned between ground heat flux and sensible and latent heat fluxes. The present model, without the advection processes, is used to study the diurnal variations of surface energy budget in both the urban (land use type C1, R2, R3) and rural (A2, A3, A4) environments. For this simulation, initial meteoro-

logical conditions and physical parameters for St. Louis, on 18 August 1976, which represents a typical clear summer day, are used. Figures 3 and 4 show contrasts in the surface energy budgets of urban and rural surfaces. Other surface parameters (T~ and w~) and mixed layer thermodynamic variables (h, ®m and Qm) are also compared for the two cases. Because of the lower albedo, urban area shows higher net solar radiation. Also, the net longwave radiation at the urban surface is smaller than that for the rural case, providing much larger (about 100 W m-2 larger) net radiation for use at the urban surface. Note that the atmospheric turbidity difference between rural and urban areas is not taken into account in the present model.

At the rural surface, more than half of the available radiation energy is used to evaporate the surface moisture in the form of latent heat flux and the remainder is divided into sensible heat flux and ground heat flux. The patterns of rural latent and sensible heat fluxes, as well as net radiation show almost symmetric shapes around solar noon. How- ever, the combination of the slight asymmetries in the same results in an asymmetric pattern of ground heat flux. The maximum in ground heat flux leads the maxima in other energy components by about 1.5 h. This partly explains the delayed occurrence of the maximum surface temperature for the rural area at 2:40 p.m. Because of the continuously available surface water to evaporate, mostly by vegetation, the Bowen ratio (fl=Hs/LE) for the rural case increases only slightly till the time of the surface maximum temperature and after that it drops sharply. The rural surface dries slowly during the day and stops drying in the late afternoon. The mixed layer specific humidity increases during early morning and remains fairly constant throughout the afternoon.

The diurnal pattern of urban surface energy balance is characterized by higher sensible heat flux and smaller latent heat flux than those of rural surface, except early in the morning. The difference in atmospheric and terrestrial longwave radiations for urban case is very small compared to that for rural and suburban cases. The ground heat flux is much larger than that of the rural case, but the larger thermal inertia of the urban subsurface medium keeps the surface temperature from rising rapidly. Still, the urban area experiences larger surface temperature and mixed layer height as compared to the rural area. The available moisture at the surface is used up rapidly in the morning as the net radiation increases and levels off in the afternoon. The Bowen ratio is much larger than 1 most of the daytime, except early in the morning. The simulated values match qualitatively with the reported Bowen ratios flu = 1.96 and fir = 0.62 for St. Louis urban and rural areas, respectively (Ching et al., 1983). Kerschgens and Hacker's (1985) empirical formula for the city of Bonn also predicts fl over 2.0 when green space is less than 10 per cent and fl = 0.6 when green space is over 50 per cent.


700

600

500

Wlm 2 400

300

20O

100 0.0

-2 -I a) P = 0.00005 Kg m s

HsCmax)

LE(max)

I I I I

O.l 0.2 0.3 0.4 Initial Surface Soil Water Content

"~ 13

.o 12

o .~ 11 O E

( b ) P = 0.00005

I0

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Ts(max)

• #- Ha(max)

• 11- LE(max)

~ - O (max) 1

K

50

4O

30

20

I0

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"0- Max. Surface Temp.

0.1 0.2

Initial Soil Water Content

1200

llOO

10o0 ~.

900

800 ~

~oo "3

60O

Fig. 2. Sensitivity of surface energy budget and mixed layer height to initial surface soil water content: (a) maximum values of energy budget components; (b) local times of maximum values of each energy budget components; (c) maximum mixed layer height

and surface temperature.

500 0.3 0.4

W/m

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1100

10

00

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8~

7

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600

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4

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RURAL

RA

DIA

TIO

N

BU

DG

ET

f "\h

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I I

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I I

I I

9 10

11

12

13

14

15

16

17

18

LOCAL TIM

E (HR)

--

R

s

----

-R

N

----

° R

. L

R A

320

315

310

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30S

300

295

290

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Sfc. Temperature

N

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eigh

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11oo

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0

400

300

200

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1200

1100

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00

900

800

700

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m 2

60

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0 40

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300

200

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NERGY B

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11

12

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14

15

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3.

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900

800

700

600

500

400

300

200

100 0

URBAN RADIATION BUDGET

(a) I

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14

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5. DYNAMICS OF THE URBAN MIXED LAYER

5.1. Balanced wind

The dynamics of an urban mixed layer are quite different from those of the homogeneous rural boundary layer. The existence of horizontal inhomogeneities in temperature and in surface roughness complicates the force balance in a stationary wind field. Here, we compare the layer-averaged stationary wind in the horizontally-homogeneous rural case with the balanced wind of the nonhomogeneous urban case. Figure 5 shows the schematics of the force balances for rural and urban boundary layers, respectively. For the urban case, the force balance diagram is modified in such a way that the resultant (V~T) of local thermal wind (VT) and large scale surface geostrophic wind (Vg) acts the same role as geostrophic wind in the rural case. From the governing equations for momentum and mixed layer height, an equation for the total mixed-layer momentum, ~=hVm, can be readily de- rived for the nonentrainment case:

~ + V'(Vm~)=g'hVtl + llh2VOm - fhk

x (Vm-- Vsm)-- Cd IVmlV m (23)

where Vm=iUm+jV m and #=9/(2®0). Stationary flow would results if conservation of momentum ¢ is assumed in the horizontal domain; i.e. when L.H.S. of the Equation (23) becomes zero. The mixed layer flow, which satisfies this simplified force balance, is called the mixed layer balanced wind. In an urban environment, effect of horizontal inhomogeneity may be very important in that the thermal winds due to spatial variations in mixed layer height and temperature must be considered. The set of balanced wind component equations for the horizontally nonhomogeneous surface can be expressed as:

I~, ÷ fh ITm-- Call Vm] ~7 m =0 (24)

ly--.fh(U m -- Gin)--Cdl Vml I7 m = 0

with l ~ = - y ' h ~ x + # h ~ :

ly = -- g'h ~ + flh 2 ~@)m cy 8y

(a) R u r a l W i n d (b) U r b a n W i n d

. f £ × v .f£× v+

^ f k x V u ^

f k × V r

Fig. 5. Schematics of (a)rural and (b)urban balanced winds, respectively.

I x and ly represent contributions due to nonhomogeneity of q and ®m in the momentum balance equation, and Gm is the layer-averaged geostrophic wind speed. No entrainment is considered in this simple case study. The ' ~ ' sign on top of a letter denotes that the variable is a component of vector in a coordinate system with the x-axis parallel to the mixed layer- averaged geostrophic wind.

When the terms due to horizontal inhomogeneity can be ignored ( lx=ly=0) , e.g. for the horizontally homogeneous rural mixed layer, the analytic solution for the stationary wind is given in the paper by Byun and Arya (1986). The balanced mean mixed-layer wind speed and cross-isobaric angle for a homogeneous rural surface are given as:

i V r l = ~ [ _ l +(1 2 1 / 2 1 /2 ÷4Emr ) ] (25)

1 t a n ( a m , ) = ~ [ - 1 +(1 + 4E2,)1/1] 1/2 (26)

,/2 where Emr = CaGm/fh is the geostrophic Ekman number for the rural mixed layer and the subscript r denotes rural surface.

It is somewhat tricky to find the analytical solution of Equation (24) for the balanced wind in the nonhomogeneous urban case. Following Byun and Arya (1986), we recognize that the stationary wind represented by the urban momentum balance equation is the same for quadratic and linear friction relations when the linear friction factor is chosen as ru=lVmu I [Cd/h]u. The analytical solution for the two wind components, for the linear friction parameterization are given as follows:

[ f ( ly~ f i x \ ] L \hJu+rUkhJu~ +f2 Gm

~.Tmu f2 + r 2

[ru(~)u--f(~)u]~-rufam ITmu--

f2 + rE (27)

where the subscript u denotes the urban case. Substi- tuting for r u = IVmul [Cd/h]u into the above equations and solving for urban balanced wind components give the following analytical results for the quadratic friction model:

fh 12 IVmul = ~ , / (28)

,/2 (~,,/2)1/2(b + G m ) - a

tan(amu) = (29) (b + Gm) + (e./2)'/2a

where a=(Ix/fh),, b=(ISfh)u and r u = { - l + [1 +4EZ~u]l/2}. The urban mixed layer Ekman number Emu is defined as:

Emu-- ~ x f a2 + ( b + G m ) 2 . (30)

AE(A) 24:4-I


For the horizontally homogeneous case, the defini- tion matches to that of the geostrophic Ekman number for the rural mixed layer. Notice that [a 2 +(b + Gm)2] 1/2 ~ IVgTI now replaces G m in rural case. We term Vg~ as the 'local thermal-geostrophic wind' (the resultant of geostrophic wind and thermal wind). The Ekman number is still useful for inferring the flow characteristics of an urban area as it varies with the spatially varying thermal wind. However, unlike in the rural case, neither the geostrophic wind speed nor [VgT[ is necessarily a good scaling velocity for the urban balanced wind. Rather, the factor [ fh /Cd] seems to be a better scaling velocity as it may not change much over an urban area since both h and C d increases in going toward the city center. The balanced wind speed normalized with the new velocity scale [ fh /Cd] is found to be a monotonic increasing nonlin- ear function of the mixed layer Ekman number (see Fig. 6). It shows that the ratio IVmul/IVg~l decreases with increasing Emu or Iv~TI. If the balanced wind is normalized with the locally modified geostrophic wind (V~T), the functional form in terms of the Ekman number is exactly the same as for the rural case shown in Byun and Arya (1986). However, the behavior of cross-isobaric angle is greatly altered from that for the rural case by the thermal wind.

5.2. Balanced wind field of a simulated urban mixed layer

The analytical solutions for the balanced wind components given in the previous section are used here to study the urban-rural differences of the day time mixed layer. To understand the underlying mechanism of the urban balanced wind analytically, we prescribe the urban heat island and the mixed layer depth in the following analytical form:

9 " , r19r, o+Aosinc2(r), 0~<r~<l (31) m~ x , y )= I

L19mo r> 1

19re(X, y)-- 19mo h(x, y) = h o + (32)

Y

m

1.0

0.6

0o?,t / / /

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

E mu

Fig. 6. Dependence of the normalized urban mixed layer balanced wind on the mixed layer Ekman

number.

Here, r = [(x /R)2+ (y/R) 2 ] 1/2 is the normalized radial distance from the origin of the coordinates located at the center of the urban area with x-axis parallel to the geostrophic wind, sinc(r) function is defined as [sin(r)/r], and R is the radius of the urban area which is assumed to be circular. In this section, Omo and h o represent rural potential temperature and mixed layer height, respectively, and Ao represents the difference in the mixed-layer potential temperatures at the urban center and surrounding rural area. The functional form of drag coefficient is assumed to be the same as that for temperature but with coefficients Cdo and Acd. The warmer mixed layer temperature in the city is a result of higher heat flux at the urban surface. The function sincZ(r) is chosen for the dome shape as it gives continuous first order derivative at r = 1, outer boundary of the urban area, as well as a smooth peak at the urban center (r=0). The expression for the urban mixed layer height (Equation 32) is based on a zeroth-order entrainment model with a constant temperature jump. The simple urban mixed layer model may help us to understand the effects of thermal gradient and surface roughness change (here represented by the change in the value of Cd) on the wind field in and around a city.

5.2.1. Effects of stability of the free atmosphere and strength of capping inversion. First, we consider the influence of conditions above the UBL, such as the stability of the free atmosphere (~) and the strength of capping inversion, which is represented by a temperature jump (A®) across the interface between the mixed layer and the free atmosphere above. Figure 7a shows an example of urban flow field given by the balanced wind equation when the inversion strength parameter I o = (2A®/~h) < 1. Table 3 summarizes the model parameters used in the flow computation. For convenience, geostrophic wind is set to be westerly. We consider this as the reference case for studying the response of urban balanced flow to changes in certain meteorological parameters.

In Fig. 7a, higher wind speed region appears to the south of the city center and lower wind speed area is located to the north. The area of increased wind speed is larger than the reduced wind speed area. Wind flow over the former experiences cyclonic turning when it enters the urban area and anticyclonic turning when it goes out of the city. On the contrary, the wind flow over the lower wind speed region shows anticyclonic turning when it enters the urban area and cyclonic turning when it goes out. Therefore, a convergence zone appears over the downwind side of the lower wind speed area. This feature is consistent with the results of previous numerical results (e.g. Vukovich et al., 1976) and observations in the St. Louis during the METROMEX (Auer, 1981) and RAMS (Shrettler, 1979) projects. Braham (1981) postulated this BL flow convergence as the major cause of the increased urban precipitation over the downwind edge of the city.

When we have a stronger capping inversion, or a less stable free atmosphere, or both, which satisfies the


(a)

- U . . . . .

I IOKM

(b)

I I IOKM

(o) | i ' '

I I 10KM

Fig. 7. Balanced wind field in and around the model urban area (gray circle at the center) Isotachs are drawn with intervals of 1 m s - 1, H represents local maximum and L represents local minimum wind speed: (a) reference case with parametric values given in Table 3, so that I o =0.5; (b) the case of hydrodynamic pressure gradient force dominating over thermal effects, with parametric values same as in the reference case, except for A® =4.0 K and ~,, =0.0075 K m- 1 so that Io= 1.33; (c)the case of no heat island (A0=0K) and much rougher urban area (Ace=0.01) with other

parametric values same as in the reference case.

condi t ion I o > 1, the flow pa t te rn over the model city changes dramat ical ly as shown in Fig. 7b. The loc- at ions of high and low wind speed areas relative to the city center are reversed• In general, the flow experiences cyclonic tu rn ing going into the city and anticyclonic tu rn ing coming out of the city. Such wind direct ion shifts of 10-20 ° have also been observed over large cities and can be explained in terms of the dependence of the cross- isobar angle on surface roughness and possibly o ther parameters . A weak

convergence zone is located over the upwind half of the high speed area and there is b road region of divergence over the downwind par t of the city.

It has been shown tha t changes in the mixed layer thermal parameters can influence the balanced flow field considerably. A plausible mechanism is described as follows. The larger tempera ture j u m p and smaller stabili ty of the upper free a tmosphere lead to higher pressure gradient force term while the s t ronger r u r a l - u r b a n tempera ture difference increases the mag-

842 D.W. BYUN and S. P.S. ARYA

Table 3. Summary of prescribed parametric values for analytical urban mixed layer model

Parameter Value Parameter Value

ho 800 m Cd0 0.005 AO 2°C Acd 0.005 7 0.01oc m T M 1 Ao 1.0oc Omo 300K G 10.0 ms -1 R 20 km f 0.0001 s- ]

nitude of the thermal wind term in Equation (23) with an opposite sign. These two opposing processes, to- gether with the imposed large scale geostrophic wind, determine the flow field around the model city. When the effect of the local pressure gradient force caused by the dome shaped capping inversion overcomes the inward thermal winds due to the warmer city temperature, the perturbation flow is outward from the center of the city, reversing the regions of high and low wind speeds. However, this situation can only be achieved for somewhat extreme values of the upper air stability and the temperature jump at the inversion base.

5.2.2. Effect of increased urban surface drag coefficient. The increased surface drag coefficient over an urban area results from larger dynamic resistance of the urban area to air flow due to increased surface roughness. By excluding the thermal wind effects of the urban heat island, the sole influence of the increased surface drag coefficient on the urban balanced flow can be studied. Thermodynamic effects on the urban wind field can be eliminated by choosing I o = 1 and A o = 0. Figure 7c shows an example of such a case when A¢d =0.01 per 20 km. It is evident that the inward flow shows cyclonic turning caused by the decrease in the magnitude of Coriolis acceleration in response to the decreased balanced wind speed over the city.

5.2.3. Effect of the urban heat island intensity. Urban heat island intensity, represented by A o here, and geostrophic wind speed are other important para-

meters determining the balanced flow field over an urban area. Figure 8 summarizes the influence of the heat island intensity on the positions and values of local maximum and minimum winds. For very small Ao, frictional effects dominate, so that no appreciable maximum is present. As the heat island intensity increases, the maximum wind speed increases and the minimum speed decreases (it goes to zero for A o larger than about 1.5 K). When the heat island intensity (Ao) is larger than about 1.5 K, double local minima appear and their positions separate more for larger A o. The position of maximum wind speed stays approximately 7.5 km south of the urban center.

6. CONCLUDING REMARKS

Using the mixed layer assumption, a two dimensional mesoscale numerical model is developed for the daytime UBL which undergoes strong diurnal evolution and is influenced by nonhomogeneous surface heating and drag of contrasting urban-rural land uses. The model incorporates the physical processes necessary to simulate the important urban-rural differences in surface energy budget. For the estimation of the surface temperature and soil moisture, the 'force-restore-method' suggested by Deardorff (1978) is utilized. Ground heat flux is estimated from the energy balance equation at the earth atmosphere in terface. Different land use types are parameterized through assigning appropriate proportions of concrete and grassy soil surface areas. Further, magnitu- des of soil thermal parameters are related to the soil water content. The entrainment process is parameterized in terms of surface fluxes using the turbulent kinetic energy balance equation. In turn, the surface fluxes of momentum and heat are estimated using the bulk-transfer relations, in which the drag and heat transfer coefficients are specified through similarity relations.

25"

20"

0 0.0

:~ 15.

.~ 10'

5,

Yminl

Yrnin , ~

Wmax x X x Y rain2

KxXXXX X x X

i ~ . Y max

" _ Jll~ I t . I b J lP ,=lk J Ik ~lt= ~ . I I . b , _ _ ;

0.5 1.0 1.5 2.0 2.5 3.0 3.5 Heat Island Intensity ( K )

15

5

-5

O

-15 "~

-20 ~ 4.0 . .

W, ndSo , " Minimum 11

Minimum 2 1 MaximumJ

Fig. 8. Locations and speeds for local maximum and minimum winds for various values of heat island intensity Ao (in K).


Sensitivity of the energy balance to soil water content is investigated for a homogeneous rural surface. Analysis shows that the time of maximum surface temperature is related to the initial values of the soil moisture content. Also, distinct diurnal patterns of urban and rural surface energy budgets, which are typical of some observational studies, are obtained. Important results of our sensitivity study are as follows. (1)The smaller urban surface albedo results in the larger net solar radiation at the surface. In addition, the small differences in atmospheric and terrestrial longwave radiation contributes to the larger net radiation at the urban surface. (2) The combined effects of large soil moisture and increased thermal inertia, for the same surface medium, are to decrease the amplitude of the surface temperature wave and to delay the time of maximum surface temperature. (3) The ground heat flux over the urban area is twice as large as that" of the rural surface, primarily due to larger net radiation and smaller latent heat flux at the urban surface. However, the larger urban thermal inertia partly reduces the effect of the large difference in ground heat fluxes in urban and rural settings on the surface temperature.

The dynamics of a dome-shaped urban mixed layer has been studied using a stationary balanced wind model. A heat-island-induced convergence zone, downwind of the model city, is predicted by the balanced wind equation. Model results indicate that changes in the mixed layer thermal parameters, such as stability of the free ,atmosphere and strength of capping inversion, can influence the balanced flow field considerably. For certain cases, when the effect of local pressure gradient force, caused by the dome shaped surface of capping inversion, may overcome the inward thermal winds due to warmer city temperature, the perturbation flow becomes outward from the center of the city. In response to the increased surface drag toward city, the mixed layer Ekman number increases, resulting in a larger cross-isobaric angle of the mixed layer wind over the center of the city. For small heat island intensity, the inward flow experiences cyclonic turning due to the dominating frictional effects. As the heat island intensity increases, however, frictional drag, in conjunction with geostrophic wind forcing, results in a region of increased wind speed, and one or two areas of reduced wind speeds over the city depending on the magnitude of heat island intensity.

Acknowledgement--This research has been partially sup- ported through the corporative agreements CR-811973 and CR-814346 with U.S. Environmental Protection Agency (EPA). Also, it has not been subjected to the agency's required peer and policy review and therefore does not necessarily reflect the view's of the agency and no official endorsement should be inferred.

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a two-dimensional mesoscale numerical model of an urban mixed layerâi. model formulation, surface...

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