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Lecture 3:
Nocturnal (Stably Stratified)
Boundary Layer
2
2
2
2
ZU
Zb
ZU
Zg
dZUd
NRi
;;
0
0
ggb 10
;
Stable Stratification – Ri > 0
Stable flows Richardson Number
2
2
22
Z
U
N
Z
U
dZ
bd
Z
U
z
g
Ri
RiK
K
dZUd
k
dZbd
K
dZUd
wu
wbRi
m
b
m
b
f
2
Thermally Driven Slope Flows
Reproduced from Mountain Meteorology (2000). Courtesy of Dr. Whiteman, PNNL.
Thermally Driven Valley flows
Reproduced from Mountain Meteorology (2000). Courtesy of Dr. Whiteman, PNNL.
Salt Lake City
downtown uptown
365 m
305 m
610 m
915 mradiosonde / ozonesonde
tethersonde
Bank One Building
Bank One Building
G-1
nightime buildup of pollutants
meteorological station
temperature
South Mountain
Nouth Mountain
chemistry
DOAS
ozone aerosols
~ 3 km
ADEQ
radar wind profiler
A Typical Urban Experiment
VTMX ASU Equipment
Theta profile in the valley
VTMX Measurements
2 7 4 2 7 5 2 7 6 2 7 7 2 7 8 2 7 9
T im e (Jd a y , L S T )
0
9 0
1 8 0
2 7 0
3 6 0
Win
d D
irec
tion
(Deg
.)
0
1
2
3
4
5
6
Win
d Sp
eed
(ms
)
4
8
1 2
1 6
2 0
2 4
2 8
Tem
pera
ture
(°C
)
-1 0 0 0
-6 0 0
-2 0 0
2 0 0
6 0 0
1 0 0 0
Rad
iati
ve f
lux
(W m
)
-2-1
4 .5 m
1 3 .8 6 m
L o n g w a v e o u tS h o rt w a v e in
(1 O c t) (5 O c t) 2 7 4 2 7 5 2 7 6 2 7 7 2 7 8 2 7 9
T im e (Jd a y , L S T )
0 .0
0 .3
0 .6
0 .9
1 .2
1 .5
(°C
)
0 .0
0 .3
0 .6
0 .9
1 .2
1 .5
(ms
)
0 .0
0 .3
0 .6
0 .9
1 .2
1 .5
(ms
)
0 .0
0 .3
0 .6
0 .9
1 .2
1 .5
(m
s )-1
-1-1
D S U S D S U S D S U S D S U S D S U S
u
vw
T
(1 O c t) (5 O c t)
Downslope – Field Data10/15 TS2
0
50
100
150
200
250
0 2 4 6
Wind Speed
Alt
itu
de
0:22
0:42
1:09
1:30
1:49
2:07
2:30
2:56
10/15 TS2
0
50
100
150
200
250
288 290 292 294 296 298
Potential Tem perature
Alt
itu
de
0:22
0:42
1:09
1:30
1:49
2:07
2:30
2:56
Flow Analysis
Idealized Slope Flow Analysis
0
0
0
H
Hwudns
n
w
s
u
HD wuUCbhbhss
hU
t
Uh ''222
sincos2
1
Ho wbBbhUs
EUhNbht
''2 cossin
EUUhs
Uh EU
EntrainmentCoefficient, Ri = bh/U2
XXX X
X X X XXX
Downslope flow - PulsationLinearized governing equations with neglected flux divergence and the entrainment-rate,
have oscillatory solution with the frequency
EaN~
or period
EaN~T
2
0222
2
UNat
UE
bat
U
UNat
bE22
Wind Speed
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
275.15 275.2 275.25 275.3 275.35 275.4 275.45 275.5
JDay
Win
d S
pe
ed
[m
/s]
Downslope flow - Pulsation
Down-slope Wind
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
275.15 275.25 275.35 275.45
JDay
Win
d S
pe
ed
[m
/s]
Wind Speed
Sin T=55 min T=55 min
EaN~T
2
ACS = 4.7 deg: T=20 – 50 min
SS = 1.8 deg: T=50 – 130 min
Other Observations
– the Riviera valley (Gorsel et al., ICAM/MAP proceedings, 2003)
– Cobb Mountain (Doran and Horst, JAM, 20(4), 361-364, 1981)
– Phoenix valley (Keon, Master Thesis, ASU, 1982)
– Slope and ACS sites of the VTMX campaign in Salt Lake City (Doran et al., BAMS, 83(4), 537-554).
0 40 80 120
Slope Site
ACS
Gorsel et al.
0
40
80
120
Critical wave period [min]
Obs
erve
d w
ave
peri
od [
min
] Doran and Horst
160 200 240
160
200
240
280
280
Keon
American Scientist 2004
Manin and Sawford’s (1978) Solutions(Combining with Briggs formula)
shI
32
sin073.0
3131
0
92
Isin5.2 U sB
3132
0
98
sin2.8 sBbI
0NFor
( is the slope angle, the stabilizing buoyancy flux driving the flow and s the along-slope distance measured downward, hI integral scales of katabatic layer depth, UI velocity and DbI buoyancy )
32
sin05.0 E
Flow Velocity
HD wuUCbhbhss
hU
t
Uh ''222
sincos2
1
s
UhU
s
UUh
2EU
High Ri Entrainment is Unimportant
)1966Rao&(Businger
sin 21
Hu bLU
Low Ri Entrainment is dominant
21
1
sin
E
bhU u
EUUhs
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 0.5 1 1.5 2 2.5 3
Ri
Lam
bd
a_c
High Ri Entrainment is Unimportant
21
sin Hu bLU
21
1
E
sinbhU u
Low Ri Entrainment is dominant
21
1
sin
E
bhU u
Parameterization of Vertical Mixing
z
)(zU
j
jj
j
iji
jj x
puqu
bwx
Uuu
x
qU
t
q 222
222
P B DFlux Richardson Number
j
iji
f
xU
uu
bwR
Gradient Richardson Number
2
2
zU
NRig
Diffusion Coefficients
ihi
i
j
j
imji
x
bKbu
x
U
x
UKuu
e.g..z
UKuw m
Flux versus Gradient Richardson Numbers
0
0.1
0.2
0.3
0.4
0.5
0.6
0.001 0.01 0.1 1 10 100Ri g
Ri f
Strang & Fernando
MY (1982)
VTMX (2000)
TA-6 data
Townsend (1958)
Nakashini (2001)
fRi
J. Fluid Mech. 2002
Eddy Coefficients
(
m s
)
M2
-1
g
(m s
)
R i
K
/ KH
2 -
1H
M
F ie ld e x p e rim e n t
E q u a tio n (3 .4 )
F ie ld e x p e rim e n t
E q u a tio n (3 .5 )
E q u a tio n (3 .6 )
F ie ld e x p e rim e n t
S tra n g & F e rn a n d o
0 .0 1 0 .1 1 1 0 1 0 0
1 0
1
0 .1
0 .0 1
0 .0 0 1
1 0
1
0 .1
0 .0 1
0 .0 0 1
1 0
1
0 .1
0 .0 1
0 .0 0 1K
K
22.0)45.0( gm RiK
45.0)07.0(
gh RiK
for the entire range of Rig;
for Rig < 1 and 07.0hK for Rig > 1
Normalization of the eddy coefficients in the VTMX
K
/
/ |d
V/d
z|
g
M
w2~
R i
K
/
/ |d
V/d
z|
Hw2
~
F ie ld E x p e rim en t
E q u a tio n (3 .7 )
F ie ld ex p e rim en t
E q u a tio n (3 .8 )
0 .0 1 0 .1 1 1 0 1 0 0
1 0
1
0 .1
0 .0 1
0 .0 0 1
1 0
1
0 .1
0 .0 1
0 .0 0 1
(
)(
)
J. Atmospheric Sci., 2003
Eddy Diffusivity (Semi Empirical)
34.0)34.0(/
~/
02.0
2
g
w
m RidzVd
K
5.049.0
2)08.0()08.0(
/~
/
gg
w
h RiRidzVd
K
45.0)07.0(
gh RiK
22.0)45.0( gm RiK
for Rig < 1 and
07.0hK for Rig > 1
Eddy Diffusivity Ratio
gRi
K /
KH
M
0.01 0.1 1 10 100
10
1
0.1
0.01
0.001
M onti et al (2002)Strang & Fernando (2001)
M R F
G S
Inverse Prandtl Number
gRi
K /
KH
M
0.01 0.1 1 10 100
10
1
0.1
0.01
0.001
M onti et al (2002)Strang & Fernando (2001)
M R F
G S
Inverse Prandtl Number
• J. Physical Oceanography 2001• Boundary layer Meteor. 2005
CROSS SECTION SW-NE 45 deg.
Temperature & Wind comparison
0
1
2
3
4
5
Win
d Sp
eed
(m s
)
-1 8 0
-1 2 0
-6 0
0
6 0
1 2 0
1 8 0
Win
d D
irec
tion
(Deg
.)
T im e (L S T )
2 8 8
2 9 2
2 9 6
3 0 0
3 0 4
3 0 8
Tem
pera
ture
(K
)-1
0 0 0 0 0 3 0 0 0 6 0 0 0 9 0 0 1 2 0 0 1 6 0 0 1 9 0 0 2 1 0 0 0 0 0 0
(a )
(b )
(c )
0
1
2
3
E (
m s
)2
-2
(d )
(averaged over 1-(averaged over 1-h, at 10 km inland h, at 10 km inland
versus versus simulations) simulations) RAMS uses RAMS uses Therry and Therry and Lacarrere’s Lacarrere’s
(1983) (1983) parameterization parameterization
(200x200 km (200x200 km domain, including domain, including
RomeRome))
Entrainment -- Encroachment of nearby fluid across a boundary
Boundary entrainment velocity (rate of propagation of a bounding surface due to turbulence).
U
uE eEntrainment Coefficient
U Characteristic velocity
Due to a normal mean flow
Flux entrainment velocity (characteristic velocity the scales cross across an interface – boundary stationary).
Downslope flow - Entrainment
x U
z
h(x)
U(z,x)
UE
E
ue
eu
eu
eu
(z,x)
TS4
TS3
TS2
TS1
x
Uhue
u
uE e
Entrainment coefficient Richardson number
2'
u
hgRi
gg 'EUUu
Entrainment Velocity
Ellison and Turner, JFM, 1959
Ellison & Turner Results
Ri
RiE
51
1.008.0
OquirrhMountain
EUUhs
ASU Doppler Lidar
ENTRAINMENT
Entrainment Coefficient
y = 0.054x-0.7494
R2 = 0.7666
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Ri
E
field
laboratory
J. Fluid Mech. 2005,
Mixing Transition -- above a certain critical Reynolds number, entrainment increases
y = 0.0532x-0.495
y = 0.0663x-0.6586
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Ri
E
Turner
Dumitrescu
Princevac
Power (Dumitrescu)
Power (Princevac)
Re vs. E
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0 500 1000 1500 2000 2500 3000
Re
E
Re big
Re small
Linear (Re big)
Linear (Re small)
Hydraulic Adjustment
Steady state, small angle
22 UCds
dhbhsinbhhU
ds
dD
ical supercritis
E
sin
bh
UF
E
sinbhU
*u
*u
5
21
21
Ri < 1
bh
UF
FCFds
dhD
221
Hydraulic Equation
050
10
13
.
~C
F when,C Critical
D
D
tical supercriis
Lh
sin
bh
UF
H
*u
2
21
Ri > 1
H''D wuUCsinbhcosbh
ss
hU
t
Uh
22
2
2
1
a) α= (10˚, 20˚) b) α= (0˚, 26˚)
Applications
Power plant emissions
Phoenix Terrain
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
7/22/19940:00
7/23/19940:00
7/24/19940:00
7/25/19940:00
7/26/19940:00
7/27/19940:00
Date & Time
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
Ozone Particles
Dispersion of Air Pollutants