A parameterization of heat and momentum fluxes in ... - LPAS - EPFL

A parameterization of heat and momentum fluxes in urban areas for
mesoscale models
Alberto Martilli * , Alain Clappier * and Mathias W. Rotach **
* Swiss Federal Institute of Technology of Lausanne (EPFL), Laboratory of Air and Soil Pollution (LPAS), CH
1015 Lausanne, SWITZERLAND, email: alberto.martilli@epfl.ch
** Swiss Federal Institute of Technology of Zurich (ETHZ), Institute for Climate Research, CH 8057 Zurich,
SWITZERLAND
Abstract
In the development of mesoscale circulation in a city area, the interactions between the urban
morphology and the geophysical characteristics of the surroundings are very important. In
this context a good representation of the heat and momentum fluxes in urban areas is crucial
for mesoscale models.
For these reasons, in this work, a modification of the surface layer parameterisations was
realised in the mesoscale model FVM (Finite Volume Model). Three active surfaces are
considered: the roofs, the walls and the canyon floor. For the exchange of momentum, two
different roughness lengths are defined for roof and canyon floors, respectively while the
contribution of the walls is parameterised with a drag force approach. The sensible heat
fluxes are determined as a function of the difference between the air temperature and the
corresponding surface temperatures. A complete energy budget equation is solved for each of
the three surfaces. The short and long wave radiative fluxes are computed by taking into
account the shadows and multiple reflection effects of the street canyon element.
The model and the parameterisation are tested for 2D idealised cases of an urban boundary
layer. A qualitative (and quantitative) comparison with measurements (momentum fluxes, heat
storage, turbulent kinetic energy profiles, etc) is performed and an analysis of the relative
impact of the three contributions is done.
1. Introduction
Air pollution problem is linked with the emission of primary pollutants, mainly in cities (NO,
VOC, SO2, CO, particulate matters) but also in rural areas (biogenic VOC), which are
dispersed by mean wind and turbulent transport and transformed, via chemical reactions that
can involve also solar radiation, in secondary pollutants (O3, PAN, particulate matters). It is
clear from this fact that two scales are involved: an ‘urban’ scale of few kilometres (the city
size), where pollutants are emitted, and a ‘meso’-scale of few hundreds of kilometres where
the secondary pollutants are formed. Since these phenomena (in particular the secondary
pollutant formation) are highly non-linear, numerical models are often used to study air
pollution problems and to help in the design of the best abatement strategies. The
meteorological fields (wind, temperature, turbulent coefficients etc.) needed for the pollutant
dispersion computation can be interpolated from measurements or computed with a numerical
dynamical model. These numerical tools should be able to reproduce the most important
mesoscale phenomena (e.g. sea/land breezes, slope winds, mountains/valley winds) and, at the
same time, have a good representation of the urban effects. Since the domain size of the
simulations is of the order of 100 Km (mesoscale), for computational reasons the horizontal
resolution cannot be smaller than 1 Km and, consequently, it is not possible to resolve
explicitly every building of the town, but their effects should be parameterised.

In this contribution a parameterisation of the urban effects in mesoscale models is proposed.
The formulation is introduced in a mesoscale model and the results are analysed on a 2D
simple case.
2. Mesoscale Model description
The mesoscale model used in this study (FVM, Finite Volume Model) solves the momentum,
mass balance, energy and moisture equations in non-hydrostatic, anelastic, Boussinesq form.
Numerically, the finite volume technique is used on a centred grid and the advection is
computed with a third order scheme (PPM, Colella and Woodward, 1984). The turbulence
scheme is k-l in the formulation of Bougeault and Lacarrere, 1989. Surface fluxes in rural
areas are computed with the Louis (1979) scheme. The ‘rural’ surface temperature and
moisture are computed with a soil module based on Tremback and Kessler 1985.
3. Urban parameterisation
Since the aim of the simulations is to reproduce and understand the dispersion of pollutants
(which are in main part emitted at ground level), the model domain starts at the street level
and the city is represented as a series of buildings of the same size, located at the same
distance one from the others (street size) and of the same height. Depending on the building
height and the vertical model resolution, several grid levels can be below the roof height.
The method consists in introducing in the momentum, energy and TKE equations solved by
the mesoscale model, extra terms representing the impact of the buildings surfaces (roof, walls
and street).
For every grid cell it is possible to define several street directions and the results will be
averaged over all the street directions in the grid cell.
The modifications are tested in a 2D simple case, with flat terrain and an idealised city
(building height 25m, street size 25m, street direction North-South, orthogonal to the plane of
the figures) of 10km surrounded by a rural area. The horizontal resolution is 1Km and the
vertical resolution ranges from 5m near the ground up to 1000 m at the top of the domain. The
geostrophical wind is 2m/s from West (left in the figures).
3.1Momentum
Three new terms are added in the equation; M M M
W i
, , , for wall, street and roof
Si Ri
respectively:
∂ρU
∂t
∂ρUU
∂ρuu
∂
ρ θ i j P ′
=− − − − gδ
3
+ M + M + M
∂x
∂x
∂x
θ
i i j
j
j i o
i W S R
i i i
The equation for the vertical velocity is not modified.
Since the street is a horizontal surface, the classical laws of the Monin Obukhov Similarity
Theory are used (in the formulation of Louis, 1979, which is more advantageous from a
computational point of view since it avoids numerical iterations). A friction velocity is
computed from a street roughness length (equal to 0.01 m in this case), mean wind speed and
atmospheric stability (difference between the air temperature in the cell and the street surface
temperature).
M
Si
2
=−ρ u *
S
U ⎡ S ⎤
i
S
2 2 ⎢
U + U ⎣V
−V
⎥
b ⎦
x
y

Where S S
is the total street surface presented in the cell, V is the volume of the cell and V b
is
the volume of the cell occupied by buildings. This contribution is different from zero, of
course, only at the first model level.
As the street, also the roof is considered as a horizontal surface. In the same way it is possible
to compute a value of the friction velocity.
2 Ui
⎡SR
⎤
M =−ρ u
Ri
* R 2 2
U U V
x
+ ⎣
⎢
⎦
⎥
y
S R
is the total surface of the roofs in the cell.
This modification is applied only at the roof level.
The sink of momentum due to building’s wall is parameterised as a drag force. This approach
is very common in modelling the impact of obstacles (for example it is used for vegetation
canopy flows, see Wilson and Shaw 1977, or Yamada, 1982, and many others). For a city, this
methodology was used by Uno et al. 1989.
M =−ρC U U
i
w drag ort i
⎡ S ⎤
W
⎢
⎣V
−V
⎥
b ⎦
U ort
is the mean horizontal speed orthogonal to the street direction, S W
is the total wall
surface in the cell. The value of the constant C drag
was fixed at 0.4 following the measurement
made by Raupach (1992) in wind tunnel for cube arrays.
Figure 1: Vertical profile computed by the model of the ratio between the local ustar and the mean wind at
different hours of the day in the centre of the urban area (h is the building height, 25m).

In Fig. 1 a vertical profile of the ratio between the local u *
(defined as
u*( z) = uw() z + vw()
z
2 2 ) and the mean horizontal wind computed by the model at
different hours in the centre of the urban area is presented
A plot of the same variable measured for a series of field studies (from Roth, 2000) is in Fig.
2. The magnitude of the calculated values is comparable with the measurements and also the
profiles have the same behaviour above the roof level. It is interesting to note, also, that below
roof level, there is a decrease of this ratio in connection with the decrease of the Reynolds
Stress as it was measured, for example, by Rotach 1993 in Zurich (not shown).
Figure 2: Vertical profiles of the ratio between local ustar and horizontal mean wind for different ‘urban’
dataset. From Roth, 2000..
During day and night, walls have the biggest impact on the sink of momentum.
The vertical section of the horizontal wind speed (Fig. 3) shows a deceleration of the flow
over the urban areas, close to the surface, and acceleration above, due to the heat island
effects. During night, the effects of the town are a reduction of the wind speed. Both
phenomena are ‘qualitatively’ in agreement with many observations recorded in urban areas.

Figure 3: Vertical section of the east-west component (parallel to the plane) of the mean wind at noon (left) and
midnight (right). The thick line on the x-axis represent the urban area.
3.2 Energy
The energy equation is modified in the following way
∂ρθ ∂ρθUi
∂ρϑui
=− − + H + H + H
∂t
∂x
∂x
i
i
W S R
As it is done for momentum it is possible to compute a value for ϑ *
for the street.
H
⎡ S ⎤
S
=−ρu
ϑ ⎢
⎣V
−V
⎥
b ⎦
S * S * S
and this is applied only at surface level.
From the value ofϑ *
computed for the roof
H
=−ρu
R * R * R
⎡SR
⎤
ϑ
⎣
⎢ V ⎦
⎥
this modification is applied at roof level.
For the heat fluxes from the walls the formulation from Clarke, 1985 was used.
H
w
hc⋅( θ
air
−θ
wall
)
=
C
with
⎛ ⎛
hc = 5. 678⋅ ⎜109 . + 0.
23⋅⎜
⎜ ⎜
⎝ ⎝
p
U
x
+ U
2 2
0.
3048
and C p
is the specific heat of the air.
y
⎞⎞
⎟⎟
⎟⎟
⎠⎠

To compute the surface temperature of roof, walls and street an energy budget equation is
solved for every surface. The radiation needed for the budget is computed taking into account
the shadow effects and multiple reflections in the street canyon.
In Fig. 4 a comparison between the total heat storage term computed by the model and the
values estimated with the Objective Hysteresis Model of Grimmond and Oke (which is a
simple parameterisation based on measurements of the storage term in urban areas as a
function of the total radiation) is presented. As it is possible to see the agreement is
particularly good.
Figure 4: Comparison between the total heat storage computed by the model in the urban area (solid line) and
that deduced from the parameterisation of Grimmond and Oke (crosses) for a three-day simulation.
The analysis of the relative importance of the different contributions shows that during day the
fluxes from horizontal surfaces, receiving more solar energy, are bigger than wall’s fluxes.
During night, on the other hand, walls are more active than street and roofs. This means that
the radiative trapping plays an important role in the heat island phenomena.
The vertical section of the potential temperature (Fig. 5) shows a typical heat island effects;
during day (the air above the city is hotter and the Boundary Layer is higher than in the rural
area) and during night the atmosphere over the city is nearly neutral and not very stable as in
the surroundings.

Figure 5: Vertical section of the potential temperature field computed by the model at noon (left) and midnight
(right).
3.3 Turbulent Kinetic Energy
In the prognostic turbulent kinetic energy (TKE) equation (formulation of Bougeault and
Lacarrere, 1989) three new terms EW, ES, ERare added
∂ρe
∂ρUe
i
∂ρue
i
=− − + ρK
∂t
∂x
∂x
32 /
i
i
m
⎡⎛
∂U
⎢⎜
⎝
⎣⎢
∂z
x
2
⎞ U
y g ∂∂ ϑ ρ ∂ϑ
⎟ + ⎛ 2
Kh
⎠ ⎝ ⎜ ⎞ ⎤
⎟ ⎥ −
z ⎠
⎦⎥
o
∂ z
− ρC e ε
+ E + E + E
W S R
lε
At the first level the entire equation is solved with a surface flux equal to zero at ground.
The impact of the street on the TKE is modelled in this way
E
S
3
⎛ u g ⎞
* S
⎡V −V
u
b
⎤
= ρ⎜
− * Sϑ
* S⎟
⎝ vk ⋅ dz / 2 ϑ
⎢
o ⎠ ⎣V
−V
⎥
b ⎦
The first term represents the shear term deduced from surface layer theory using the value of
the street friction velocity. The second term is the buoyancy term based on the street surface
heat fluxes. Both terms are volumetric. Again, this term is different than zero only at the first
level. dz is the height of the grid cell.
Similarly for the roof:
E
R
3
⎛ u*
R
g
= ρ⎜
− u
⎝ vk ⋅ dz / 2 ϑ
o
ϑ
* R * R
⎞ ⎡ Vb
⎟ ⎢
⎠ ⎣V
−V
this term is different than zero only at roof height.
b
⎤
⎥
⎦

The impact of the walls is modelled in a way similar to what it is done in vegetation canopy
models:
3
w drag ort
E = ρC U
⎡ SW
⎢
⎣V
−V
b
⎤
⎥
⎦
In the formulation by Bougeault and Lacarrere, the length scales used for the computation of
the dissipation and that for the turbulent diffusion coefficients, are function of the TKE value
in the point and the profile of potential temperature.
In the case of an urban surface, the presence of buildings can induce some circulation of a size
comparable with that of the roughness element. For this reason, below roof level we added a
second length scale (the roughness element size):
1 1 1
= +
l l hu
B
where hu is the building’s height and l B
is the length scale computed with the original
formulation.
Vertical profiles of TKE normalised by the square root of the Reynolds Stress at the roof
height are shown in Fig. 6.
Figure 6: Vertical profile of the ratio between the turbulent kinetic energy and the square root of the Reynolds
Stress computed by the model in the centre of the urban area at different time of the day.
Those values are consistent with the measurements of Rotach (1993) around roof height and
with that of Feigenwinter et al. at 2-3 times the building height.
As for momentum, also for TKE the most active surface is the walls.
4. Conclusions
A parameterisation for heat and momentum fluxes for mesoscale models in urban areas is
presented. The formulation takes into account separately the impact of the three active
surfaces of the roughness element: wall, street and roof. The modifications were introduced in
a mesoscale model and tested in a simple 2D case. A comparison with series of field

measurements shows that the new formulation is able to reproduce the main features of
momentum, heat and turbulent fluxes in urban areas.
From the analysis of the impact of the different terms it is possible to see that the walls are the
most active surfaces for momentum and TKE during the day and during the night. For heat, on
the other hand, street and roofs are more active during daytime and walls during night-time.
Acknowledgements
The authors would like to thank M. Roth for provides Fig. 2.
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