Lecture 2 Atmospheric Boundary Layer Neutral, Convective, Stable and Transitional Boundary Layers Region of the lower atmosphere where effects of the Earth surface are felt Surface – fluxes of momentum, buoyancy….. H. J. Fernando Arizona State University
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Lecture 2
Atmospheric Boundary Layer
Neutral, Convective, Stable and
Transitional Boundary Layers
Region of the lower atmosphere where effects
of the Earth surface are felt
Surface – fluxes of momentum, buoyancy…..
H. J. Fernando
Arizona State University
Atmospheric Boundary Layer (flat terrain)
ij
uux
U
xx
PUf
x
UU
t
Uji
j
i
jij
ij
i
1
~~
Horizontal homogeneity
Steady (Boun Layer)
3
3
~~
1
xx
PUf i
i
y
(y
low
P
High P
x
z
ze
Fig. 4.7.1 Ekman Spiral in the Northern Atmosphere
y
(y
low
P
High P
x
z
ze
Fig. 4.7.1 Ekman Spiral in the Northern Atmosphere
PUf
x
PUf
g
ig
1
1
~~
~~
Geostrophy
0=PUg
.~
3
3
xUfUf i
g
+×=×
~~
z
UUf
zvvf
yz
g
xzg
m 50≈101010
5020-1≈
≈-(1
24
22
2
××
×.~
)(
)
*
*
g
g
fv
uaH
Hfvua
Surface layer – small change of stress
y
(y
low
P
High P
x
z
ze
Ekman Spiral in the Northern Atmosphere
dzvvfHH
gxzxz 0
0
u*2
Boundary Layer fow
ig
x
PUf
1
=×
~~
3
31
xx
PUf i
i
3
3--x
UUf ig
=× )(
~~~
z
UUf
zvvf
yzg
xzg
( ) ( )0,, gUvU = at z ze
gg UUU 0v at z = 0
;'z
UKwu
z
Uxz
z
vKyz =
(K theory)
y
(y
low
P
High P
x
z
ze
Fig. 4.7.1 Ekman Spiral in the Northern Atmosphere
Ug
e
zz
g
z
zeUU e cos1
e
zz
g
z
zeUV e cos
f
Kze 2
Ekman Spiraly
(y
low
P
High
P
x
z
ze
Fig. 4.7.1 Ekman Spiral in the Northern Atmosphere
Formation and Breakdown of an Inversion Layer in El
Paso
Unstable Boundary Layer
(flat terrain)
Stable Boundary Layer (flat terrain)
Convection between horizontal surfaces
2,,,, fkHq To
2
4
0RaT
fk
Hq
k
Tdg 3
2
424
HTa
T
rk
P
,
[Rayleigh-Benard ; Chandrasekhar 1961]
( )
)(
~
~
*
*
1971
10≈
0
8
310
Deardorff
WqB
RHL
HqW
f
=
Molecular
Thermals
31RaNu
Goldstein and Chu (1973)
Sparrow et. al. (1970)
Plumes - Convective
Molecular
Thermals
31RaNu
Goldstein and Chu (1973)
Sparrow et. al. (1970)
Dave Fultz’s
experiments
Irregular vortex patterns
(Higher Ra/smaller Ta)
“Geostrophic Turbulence”puuu
1~2~.
Onset of Rotational Effects
kjjk
HH
H uRoz
uw
x
uu
t
u
Tu
L
1
x
p
u
p
H
2
0
0
2
222
z
u
L
L
xx
u
V
H
Re
1
310
1
/)(
~
HH
H
H
LqU
fL
uRo
=
=
km100~
( )213
0
210
31
10≈
2≈
/
//
)(
)(~
fqL
fqLqU
R
Ho
Sea Surface Temperature, July
-10.0 33.2 ( oC )
Non-Rotating Plume
Rotating Plume
Fernando, Dyn. Atmos. Oceans, 2000
Atmospheric Surface layer
Monin-Obukhov (1954) Similarity Theory
-- For flat terrain surface layer
pC
Qgq
0
00
=buoyancy flux
pC
QwH
0
0 ==___
)(temperature flux
Heat Flux Q0, Stress τ0 = u*2
Parameters
Q0τ0
Define the scaling variables:
0*
*
u
wT
temperature scale
0w 0* TConvection
0w 0 :stable * TStratification
1/2
0* wuu velocity scale
Monin-Obukhov scale
0
33 11
q
u
wg
uL ***
=
Θ
=
Non dimensional relations
z
u
Z
U
*
z
T
Z
*=
*u
w
*T
w
Øm (z/L*) wind shear)
Ø h (z/L*) (thermal stratification)
(variability of w)
(variability in θ)
Øw =
Øθ =
(dissipation)
z
u
3* Øε (z/L*)
),,(
),,(
**
0*
LzuG
qzuFAny
Kaimal &
Finnigan
1994
fRi
dZ
Udwu
wb
Z
u
wg
L
Z
3
**
;
given that
Z
u
dZ
Ud *
2
*uwu
*Lz shear dominates : *Lz Buoyancy (outer layer)
With a slope
Thermal blob
(I)
(IV)
(III)
(II)
(I)
(IV)
(III)
(II)
Detachment occurs when
33
10
cTgRaRa c
Princevac &
Fernando, Phys.
Fluids, 19, 2007
Convection in Complex Terrain
T-Rex Observations (NCAR)
zz
x=0
h
frontal wave
x=L
x=L
c= b h
x
x
W
E
V
I
(z)
S
SF
c
H
upslope flow, U
(x, z, t)
frontal wave
~
z (x)S
Fully developed upslope flow
Prandtl’s Solutions
000 TTgbg
zTT Γ+= 0
gNdz
bd 2
Initial temp distribution
Initial hydrostatic
sin1
00
bs
p
cossin nsz
Now give a perturbation, b’ and corresponding velocity u
2
'2
s
b
s
bu
2
2
0n
uvb += sin'
0 spandgb
sin/// 2Nszzbsb
0'sin' 22
4
4
b
v
N
n
b
nlAeb ln cos' /
41
22 sin
4
N
vl
l
ne
vNAu l
nsin
21
2
Velocity along the slope, constant (eddy?) coefficients
/0lqA
nbq /0
constant heat flux boundary condition
[S]
[M]
[I]
[E]
zh
SS
SS
z
x=0
h
U
front wave
x=L
x=L
Th
c= b h
x
x
h
W
S
I
E
S
MS
mI
V
I
S
C
(z)
C
S
S
SF
c= b hI
Upslope - Theoretical Model
z
F
zW
xU
t
zb
x
P
z
UW
x
UU
t
U
ˆˆˆ
ˆˆˆ
)(
ˆ)(
th
zftUU m
S
u
0
S
SFF
}
}0ˆˆ zz
hzz ˆˆ
*LCS
SS
..................
ˆ 31
34
*
F
z
Uhzw
..................
ˆ*
F
z
Uzu
Hunt, Fernando & Princevac
J. Atmos. Sci., 60, 2003
Arizona State University
Environmental Fluid Dynamics
Program
Theory - Up-Slope Velocity
For small
*3
1
wU uM
where
31
31
31
)( 0* hqhgFw S
4u (?)
(Experiments)
Arizona State University
Environmental Fluid Dynamics
Program
Experimental setup - Schematic
Balloons
Arizona State University
Environmental Fluid Dynamics
Program
VTMX velocity profileVTMX Velocity Profile
0
50
100
150
200
250
300
350
0 0.5 1 1.5 2 2.5 3
Velocity [m/s]
He
igh
t [m
]
10/08/00 5:53PM(Qnet = 91 W/m 2̂)
10/14/00 4:58PM(Qnet = 49 W/m 2̂)
Arizona State University
Environmental Fluid Dynamics
Program
Up-slope velocityVTMX Daily Averaged Um VS w*
1/3
(October 1 - 5, 7, 14 - 17)
(Days with low synoptic wind condition)
y = 4.1458x
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Daily Averaged w*1/3 [m/s]
Da
ily
Av
era
ge
d U
m [
m/s
]
Laboratory Data
VTMX Sonic Data
VTMX Balloon Data during IOP
Linear (VTMX Sonic Data)
*3
1
wU uM
Geophysical Convection
A continuum of scales
• Large scale -- deep convection/Hadley
Cells (~ 10000 km)
• Thunderstorms (~250mkm)
• Slope flows (10-100 km)
• Atmospheric Plumes -- Microbursts (2 km)
• CBL (100m to km)
Drivers of Environmental Motions
TidesMoon
SUN
Hadley Circulation
warm
cold
p
1
~
g
b ~~
uf
T
T
a
b
kP
dT
d
ababT
ab
Tdg
ra
a
T
;4
4
Ro
2
42
2
42
22
2
1
Atmospheric Convection
CONVECTION OVER URBAN
AREAS
Phoenix Metropolis
Urban Heat Island -- Urban air can be
significantly hotter than the countryside
UHI in satellite image of
Phoenix
This graph illustrates that rapid temperature increases in Phoenix correspond to rapid population growth. Baltimore’s population growth peaked just after 1960, which corresponds to slight temperature changes. Notice that urban effects for both cities are most dramatic in minimum temperatures
Urban-Rural Monthly Average Max and Min Temperature Differences for BES and CAP LTER