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Slab surface energy balance scheme and its application to parameterisation of the
energy fluxes on urban areas
Krzysztof Fortuniak University of Łódź, Poland
Brian OfferleGöteborg University, Göteborg, Sweden;
Indiana University, Bloomington, USA
Sue Grimmond Indiana University, Bloomington, USA
Outline
1) Urban atmosphere and urban models
2) Motivation3) Slab surface energy balance
model4) Energy balance measurements
in Lodz5) Modeled and measured urban
energy balance components6) Model applications7) Conclusions
urban boundary layer
urban canopy layer
1. Very complex models: vegetation, windows, indoor processes, etc.
2. More generalized models: simplified geometry, uniformed surfaces
3. Slab models: town is treated as a single entity with specified physical parameters
Boundary layer models: different models from 3D to 1D with different turbulence parameterizations
Surface energy balance models:
3. Slab models: town is treated as a single entity with specified physical parameters
1D model with first order turbulence closure
presented model
Urban Atmosphere and Urban Models
Motivation
Why slab approach ?• Simple – low time consuming
• Easy to link with mesoscale and GSM models
• Good for studies on the role of individual parameter
Questions:• Is a slab model able to capture
singularities of the urban energy balance components?
• Which parameters are crucial for modification of the local climate by urbanization?
The model
Radiation budget: Q* = (1-α) Itoth + εL↓ – εσTs4
Itoth - short-wave radiation on the horizontal surface after Davis at al. (1975):
3 6 9 12 15 18 210
100200300400500600700
W m
-2
18 III 1999
h
3 6 9 12 15 18 210
100200300400500600700
W m
-2
21 I 1999
h
3 6 9 12 15 18 210
100200300400500600700800900
1000
W m
-2 10 V II 1999
h
Validation of the Itoth model again Lodz data (selected sunny days)
Itoth = S0sinhs·τwa·τda(1 + τws·τds·τrs)/2
S0 - solar constant, hs - solar height, and τ - transmissions due to water vapor absorption, aerosol absorption, water vapor scattering, aerosol
scattering, and Raleyigh scattering L↓ -incoming longwave radiation: taken constan or calculated with empirical formula (e.g. Idso & Jackson, 1969)
L↓= [1 – 0.261·exp{-7.77·10-4 · (273-T)2}] σT4
The surface energy balance model: Radiation budget: Q*+QG+QH+QLE=0
Heat flux to the ground (QG) and temperature profile is found by numerical solution of the one-dimensional heat diffusion equation:
2
2
g zT
νtT
νg - thermal diffusivity
T-temperature at depth z
Numerical scheme: Crank-Nicholson
Number of levels: 10 levels
Lower boundary conditions:constant temperature
Upper boundary conditions: temporal evolution of the surface temperature
The surface energy balance model: Heat flux to the ground: Q*+QG+QH+QLE=0
Q*+QG+QH+QLE+QS=0
QS
QG
QG
Q*+QG+QH+QLE=0
0. 5cm3cm7. 5cm
1. 5cm5cm10. 5cm
15. 5cm25. 5cm
45. 5cm
95. 5cm
Validation of ground temperature calculations – comparison with temperature (5cm above ground) evolution at Lodz-Lublinek meteorological station in calm, cloudless nights (QH,QLE=0)
0 2 4 6 8 10 12t [h]
6
8
10
12
14
16
18
20
22
T [
0C
]
8/9.09.1999 C =1.3 10 -6J m -3K -1
k=0.3 W m -1K -1
L =315 W m -2
T o=21.5 oC , T G=24.4 oC T o=12 oC
0 2 4 6 8t [h]
0
2
4
6
8
10
12
T [
0 C]
23/24.05.1999 C =1.3 10 -6J m -3K -1
k=0.2 W m -1K -1
L =290 W m -2
T o=10.5 oC , T G=18.6 oC T o=8 oC
0 2 4 6 8t [h]
6
8
10
12
14
T [
0 C]
9/10.06.1999 C =1.3 10 -6J m -3K -1
k=0.4 W m -1K -1
L =315 W m -2
T o=13.5 oC , T G=18.8 oC T o=6 oC
0 2 4 6 8 10t [h]
4
6
8
10
12
14
16
18
T [
0 C]
16/17.08.1997 C =1.2 10 -6J m -3K -1
k=0.2 W m -1K -1
L =305 W m -2
T o=16.4 oC , T G=20.8 oC T o=12 oC
0 2 4 6 8 10 12 14t [h]
-14
-12
-10
-8
-6
-4
-2
T [
0 C]
25/26.01.1998 C =1.3 10 -6J m -3K -1
k=0.3 W m -1K -1
L =228 W m -2
T o=-3 .5 oC , T G=3.4 oC T o=8 oC
0 2 4 6 8 10 12t [h ]
- 2
0
2
4
6
8
1 0
1 2
1 4
T [
0 C]
30/31.03.1999 C =1.7 10 -6J m -3K -1
k=0.7 W m -1K -1
L =250 W m -2
T o=12.5 oC , T G=10.5 oC T o=10 oC
Surface cooling in calm cloudless nights Energy balance: Q*+QG+QH+QLE=0
Parametrisation of turbulent heat fluxes (QH and QLE) bases on Monin-Obukhov similarity theory with Businger’s functions for the flux-profile relationships. Method proposed by Louis (1979) with Mascar at al. (1995) modification is used.
Fuxes: and
are found from profile relationships:
Lz
Lz
zz
lnku
)z(u m0mm
m0
2'' uuw u''w
Lz
Lz
zz
lnk
R h0hh
h0
stability parameter z/L is found by iterative solution of:
Lz
Lz
zz
ln
Lz
Lz
zz
ln
RRi
Lz
h0hh
h0
2
m0mm
m0b2b u
gzRi
where Rib is the bulk
Richardson number :
In calculations of the turbulent moisture flux additional surface resistance is considered acording to Best (1998) method.
The surface energy balance model: Turbulent heat fluxes : Q*+QG+QH+QLE=0
One dimensional first order model
– 28 levels form 2m to 5000m
– constant upper boundary condition
– different local turbulence closure schemes tested ( K–l ):
o Louis (1979)o Mellor and Yamada (1982)o Gambo (1978)o Sievers and Zdunkowski (1986)
- advection estimated by simultaneous calculation for rural and urban points
The boundary layer model
5 10 15 200
500
1000
1500
2000
Modeled (dashed) versus measured (solid) temperature and humidity profiles in day 33 Wangara experiment (9.00h, 12.00h, 15.00h)
5 10 15 200
500
1000
1500
2000
5 10 15 200
500
1000
1500
2000
5 10 15 200
500
1000
1500
2000
0 0.002 0.004q[kg/kg]
0
500
1000
1500
2000
0 0.002 0.004q[kg/kg]
0
500
1000
1500
2000
0 0.002 0.004q[kg/kg]
0
500
1000
1500
2000
0 0.002 0.004q[kg/kg]
0
500
1000
1500
2000
Louis Mellor-Yamada
Gambo Sievers & Zdunkowski
temperature [C] temperature [C] temperature [C] temperature [C]
The boundary layer model – model validation
Modeled temperature and wind speed profiles over urban and rural sites in the night
1 0 1 2 1 4 1 6 1 8 2 0temperature [C]
0
100
200
300
400
high
t [m
]
urbanrural
0 1 2 3 4 5 6 7wind speed [ms- 1]
0
100
200
300
400
high
t [m
]
urbanrural
Model testing – vertical profiles
Energy balance
measurements in
Lodz
#Y#Y
#Y
1 0 1 2 3 4 5 Kilometers
N
EW
S
old town
blocks of flats
industrial Energy balance measurement point
Lodz-Lublinek
meteorologicalstation
Measurements in Lodz
Energy balance measurement point
Energy balance measurement point
Energy balance measurement point
Measured and
modeled energy
balance components
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (March 7th, 2001)
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
[Wm
-2]
Q*
Q
Q
Q
H
E
7.03.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
[Wm
-2]
Q*
Q
Q
Q
H
E
7.03.2001
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.5 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 230 Wm-1 albedo: α = 0.08; emissivity: ε = 0.9;soil moisture content: SMC = 35%
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (March 28th, 2001)
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.5 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 220 Wm-1 albedo: α = 0.13; (snow)emissivity: ε = 0.85;soil moisture content: SMC = 15%
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600[W
m-2
]Q*
Q
Q
Q
H
E
28.03.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600[W
m-2
]Q*
Q
Q
Q
H
E
28.03.2001
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (April 30th – May 3rd, 2001)
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.5 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 310 Wm-1 albedo: α = 0.08; emissivity: ε = 0.9;soil moisture content: SMC = 3%
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600
800
[Wm
-2]
Q*
Q
Q
Q
H
E
30.04-3.05.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600
800
[Wm
-2]
Q*
Q
Q
Q
H
E
30.04-3.05.2001
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (July 7th, 2001)
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.5 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 370 Wm-1 albedo: α = 0.08; emissivity: ε = 0.9;soil moisture content: SMC = 8%
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600
800
[Wm
-2]
Q*
Q
Q
Q
H
E
7.07.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600
800
[Wm
-2]
Q*
Q
Q
Q
H
E
7.07.2001
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (August 19th, 2001)
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.5 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 370 Wm-1 albedo: α = 0.08; emissivity: ε = 0.9;soil moisture content: SMC = 7%
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600[W
m-2
]
Q*
Q
Q
Q
H
E
19.08.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600[W
m-2
]
Q*
Q
Q
Q
H
E
19.08.2001
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (October 10th, 2001)
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.5 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 340 Wm-1 albedo: α = 0.08; emissivity: ε = 0.9;soil moisture content: SMC = 4%
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
[Wm
-2]
Q*
Q
Q
Q
H
E
3.10.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
[Wm
-2]
Q*
Q
Q
Q
H
E
3.10.2001
The energy balance components for the center of Lodz. Comparison of the results of measurement (thin lines) and simulation (thick lines)
Measured and modeled urban energy balance components in Lodz (December 12th, 2001)
Parameters used in simulation: ground heat capacity: Cg = 2.0 106 J m
-3 K-1; ground thermal conductivity: kg = 1.0 Wm-
1K-1; roughness length for momentum: z0m= 0.6 m; roughness length for heat: z0h = 0.00001 m;
incoming longwave radiation: 200 Wm-1 albedo: α = 0.23; (snow)emissivity: ε = 0.85;soil moisture content: SMC = 35%
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
[Wm
-2]
Q*
Q
Q
Q
H
E
8.12.2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
[Wm
-2]
Q*
Q
Q
Q
H
E
8.12.2001
The energy balance components for the center of Łódź (left) and nightly temperatures courses at a rural and urban station (right). Comparison of the
results of measurement (thin lines) and simulation (thick lines)
Modeled and measured temperature evolution
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600
800
[Wm
-2]
Q *
Q H
Q E
Q
12 15 18 21 0 3 6 9 12t [h]
10
15
20
25
30
T [
oC
]
Tu
Tr
28 July 2001 27/28 July 2001
0 3 6 9 12 15 18 21 24t [h]
-200
0
200
400
600
800
[Wm
-2]
Q *
Q H
Q E
Q
12 15 18 21 0 3 6 9 12t [h]
10
15
20
25
30
T [
oC
]
Tu
Tr
28 July 2001 27/28 July 2001
0 3 6 9 12 15 18 21 24t [h]
-200
-100
0
100
200
[Wm
-2]
12 15 18 21 0 3 6 9 12t [h]
-15
-10
-5
0
T [
oC
]
Tu
Tr
8 December 2001 7/8 December 2001
0 3 6 9 12 15 18 21 24t [h]
-200
-100
0
100
200
[Wm
-2]
12 15 18 21 0 3 6 9 12t [h]
-15
-10
-5
0
T [
oC
]
Tu
Tr
8 December 2001 7/8 December 2001
Model applications
Dependence of the urban-rural temperature differences on the distance from a city border. Curves show a logarithmic fit to
the data
Model application – UHI and population
0 5000 10000 15000 20000 25000 30000D [m ]
0
1
2
3
4
5
Tm
x [o
C]
Ug[ms-1] L [Wm-2] SMCR 3 320 0.05 5 320 0.05 3 I&J 0.05 5 I&J 0.05 3 320 0.20 5 320 0.20
∆Tmx ~ log (D )
P ~ D2
∆Tmx ~ log (P )
0 2 4 6 8 10 12 14 16U g [m s -1]
0
1
2
3
4
5
6
7
Tm
x [o
C]
0.40 0.01
0.20 0.05
0.05 0.05
0.30 0.30
SMCR SMC U
L =320 Wm -2
0 2 4 6 8 10 12 14 16U g [m s -1]
0
1
2
3
4
5
6
7
Tm
x [o
C]
0.40 0.01
0.20 0.05
0.05 0.05
0.30 0.30
SMCR SMC U
L - I&J
Modeled dependence of the UHI intensity (T) on the wind speed
Model application – UHI and wind speed
3) power
2) exp.
Function types:
1) classical va(N)
T
vb(N)ea(N)T
dc)(va(N)
T
0 2 4 6 8 10 12 14 16U g [m s -1]
0
1
2
3
4
5
6
7
Tm
x [o
C]
0.40 0.01
0.20 0.05
0.05 0.05
0.30 0.30
SMCR SMC U
L - I&J
0 2 4 6 8 10 12 14 16U g [m s -1]
0
1
2
3
4
5
6
7
Tm
x [o
C]
0.40 0.01
0.20 0.05
0.05 0.05
0.30 0.30
SMCR SMC U
L =320 Wm -2
0 2 4 6 8 10 12 14 16U g [m s -1]
0
1
2
3
4
5
6
7
Tm
x [o
C]
0.40 0.01
0.20 0.05
0.05 0.05
0.30 0.30
SMCR SMC U
L - I&J
0 2 4 6 8 10 12 14 16U g [m s -1]
0
1
2
3
4
5
6
7
Tm
x [o
C]
0.40 0.01
0.20 0.05
0.05 0.05
0.30 0.30
SMCR SMC U
L =320 Wm -2
Isotherms of the UHI intensity (ΔTmx) as a function of wind and cloudiness
Model application – UHI and wind speed
V [m/s]
N
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10
1
2
34
A
V [m/s]
N
0
1
2
3
4
5
6
7
8
1 2 3 4 5 6 7 8 9 10
1
2
34
B
V [m/s]
N
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10
1
2
34
C
V [m/s]
N
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10
1
2
34
D
A – spline functions fitted to the data from Łódź (1997-1999)
B – classical fit ∆Tmx = (3.43 -0.033N
2)∙v −0.5 Explains 58.7% of T variance
C – power fit ΔTmx = (14.9 -0.14·N
2)· ·(2.28 + v)–1.22
Explains 61.0% of T variance
D – exponential fit ΔTmx = (5.51 – 0.50·N)·
·e –(0.41–0.067·N+0.005·N·N) ·v
Explains 61.2% of T variance
Modeled nighttime temperature courses for sites which differ roughness length only. On the left plot sites with different roughness length of temperature z0h
(the same z0m=0.2); on the right plot sites with different roughness length of momentum z0m (the same z0h=0.01);
- 1 2 - 9 - 6 - 3 0 3 6 9 1 2
1 6
2 0
2 4
2 8 z 0h [m]1e- 11e- 21e- 31e- 41e- 51e- 6
- 1 2 - 9 - 6 - 3 0 3 6 9 1 2
1 6
2 0
2 4
2 8 z 0m [m]0. 010. 10. 51. 02. 04. 0
Model application – the role of roughness lengths
SUEB and UHI– the role the thermal admittance
Modeled variation of the surface temperature following sunset for materials with different thermal admittances (μ) of the ground:
1) μ=600, 2) μ=1000, 3) μ=1400, 4) μ=1800, and 5) μ=2200 J m‑2 s‑1/2 K‑1.
Different combinations of initial surface temperature (To) and temperature of deep soil (TG) selected. In all cases L↓=260 Wm‑1
0 2 4 6 8 10t [h]-8
-6
-4
-2
0
2
4
6
8
T [o C
]
1
2
3
45
0 2 4 6 8 10t [h]-8
-6
-4
-2
0
2
4
6
8
T [o C
]
1
2
3
4
5
Tsurf=7oC, Tdeep=0oC
Tsurf=7oC, Tdeep=7oC Tsurf=7oC, Tdeep=14oC
Slab models with properly chosen parameters can satisfactorily reproduce many singularities of the urban climate and can be use as a tool for investigation of the modification of a local climate by the urbanization.
Conclusions: