Analytical Solution of Cold-air-drainage Flow Within and Above ...

2 nd International Countermeasures of Urban Heat Island, Berkeley, USA, Sep21-23, 2009
Analytical Solution for Cold-airdrainage
flow On Sloping Forest
Zhiwen Luo, Yuguo Li
Department of Mechanical Engineering
The University of Hong Kong, Hong Kong, China
Chuixiang Yi
Queens College, The City University of New York,
Flushing, USA

Cold-air-drainage Flow
Source: C.David Whiteman “Mountain Meterology”

Cold-air-drainage Flow
• Mitigate the nocturnal UHI
– Kitada, 1998; Ohashi and Kida, 2002
• Disperse the urban pollution
– Baumbach and Vogt, 1999; Egan, 1984; Lu and Turco, 1994
• Influence the nocturnal ecosystematmosphere
exchange
– Lee and Hu, 2002; Turnipseed et al, 2003; Yi et al, 2000

Hong Kong Island
Source: www.gearthblog.com

Physical model

19:00 pm
Mar 15,2008

Ra=10 9 Ra=10 8
Ra=10 7
Ra=106

Analytical Models
• Prandtl model
– One dimensional, but gives the detailed structure of
the flow profile
• Hydraulic model
– Only provide the layer-averaged characteristic scales
of flow parameters, i.e., velocity, momentum
thickness, buoyancy deficit

Vegetation on Slope Flow
• Few studies address this problem
– Bergen, 1969; Devito and Miller, 1983
• Katabatic flow can occur both within and
above the tree canopies
– Komatsu et al, 2003; Devito and miller, 1983; Pypker, 2007
• Understand the flow structure can help to
estimate the surface fluxes

Cold-air-drainage winds---Forest Canopy
katabatic wind above the canopy
katabatic wind in the canopy
Modified from Prof Tree’s PPT

Aims
Propose a simple analytical model
by coupling both above and within
tree canopies

Cold-air-drainage Flow Model
z
n
h
0
x
z
s
T
Assumptions:
_
• One dimensional normal to
the slope
• Constant deficit of potential
temperature in the canopy
• Non-linear advection term is
ignored in the momentum
equation
c D --- drag coefficient;
a ---- leaf area density;
LAI—leaf area index;
"! --- deficit of the potential
temperature ;
h ---- canopy height

Flow Within Canopy
(non-uniform)
Momentum equation
" u#
w#
=
" n
g&
!%
sin$
+ c
D
au 2 ( n)
Parameterizing the Reynolds stress (Yi,
2008):
$ ( z) / # = " u!
w!
( z)
= c
D
( z)
u 2 ( z)
U
"
0
cD(0)
2 $ [ LAI $ L(
n)]
g'
#&
sin%
$ [ L(
n"
) $ L(
n)]
( n)
= $ ( U (0) e $ ! e dn"
c
( 2
D(
n)
a(
n)
u
D
2
cD ( n)
u ( n))
= g%
! $ sin#
+ c
" n
( n)
c
D
( n)
n
( n)
)
1/ 2
n
L( n)
= " a(
n!
) dn!
LAI = L(0)
# h

Flow Above Canopy
Prandtl Model
!
!
"
!
!
#
$
%
=
=
%
2
2
2
2
)sin
(
)
(
sin
dn
d
k
n
u
dn
n
u
d
k
g
h
m
&
'
(
'
&
)
)]
/
'cos(
)
/
sin(
[
)
(
/
l
n
C
l
n
Ke
n
u
s
l
n
!
"
=
!
#
4
1
2
2 )
sin
4
(
!
N
k
k
l
h
m
=
m
h
k
k
N
g
K
!
=
Prandtl
Prandtl in 1905
in 1905

Coupling at Canopy Top
canopy
above
canopy
in
u
u !
! = )
(0
(0)
s
canopy
above
n
canopy
in
n !
!
! "
=
= "
" #
=
#
= ,
0
0,
canopy
above
n
canopy
in
n
dn
du
dn
du
!
=
!
=
=
0,
0,
!
"
!
#
$
%
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
'
(
=
)
)
'
&&&&&&&&&&&&&&&
*
(
'
= '
'
*
'
'
'
'
+
0
)]
/
cos(
)
/
sin(
[
)
(
0
)
)
(
sin
)
(
(0)
(
)
(
/
2
1/
0
)]
(
)
(
[
)]
(
[
2
2
n
l
n
C
l
n
Ke
n
U
n
h
dn
e
n
c
g
e
C
K
n
c
c
n
U
s
l
n
n
n
L
n
L
D
s
n
L
LAI
D
D
,
-
,
.

Validation and Discussion
Leaf area density

Velocity Profile
Low-level jet
Ri
g ! T
= T ! z " #
2
$ ! u %
& '
( ! z )
Minimum vertical exchange
Super-stable layer
Minimum velocity and largest leaf area density

Sensitive Study
---Uniform Case
• Atmospheric Stability
• Slope angle
• Canopy morphology

Influence of Atmospheric Stability and Slope Angle
400
350
300
!=5°, "=2 K/km
!=5°, "=4 K/km
!=10°, "=2 K/km
Steeper slope
)
m
(
H
250
200
150
Weak stability
100
50
Strong stability
Canopy top
0
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
Velocity (m/s)

)
m
(
Influence of Plant Canopy
y
p
o
n
a
c
220
200
180
LAI=2
LAI=4
LAI=10
e
h
t
e
v
o
b
a
160
140
120
100
80
LAI=4
LAI=2
t
h
g
i
e
H
60
40
20
LAI=10
0
Canopy Top
-20
-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0
Velocity (m/s)

Conclusions
• Analytical solution on cold-air-drainage winds by
accounting for the influence of tree canopy is
obtained.
• The effect of atmospheric stability and slope
inclination is also investigated.
• The influence of different leaf area indexes is
studied.

Thank you!