Final Review
1. What is atmospheric boundary layer?
The lowest portion of the atmosphere (from surface to about 1 to 2 km high) that is directly affected by surface turbulent processes.
2. Taylor Hypothesis
A turbulent eddy might be considered to be frozen as it advects past a sensor.
3. Material (total) derivative
v
tzw
yv
xu
t
Dt
D
4. Statistic representation of turbulence
a. Mean and perturbation
;rrr ;
;ww w;vv v;uuu
n
i 1iN
1
The average could be temporal, spatial, or ensemble average depending on specific dataset.
b. Reynolds average
aaa)aa(a)( 0a
baabab)( 0ba)ba(
)ba()ba(
)aa(aa)(
covariance
variance
2/122/1a )a()aa( Standard deviation
ba
)ba(ab Correlation coefficient
c. Turbulent kinetic energy (TKE) )wvu( 22221
5. Turbulent flux
a. Sensible heat flux, Latent heat flux, buoyancy flux
Sensible heat flux, SH22 m
Wsm
J:unit
TuC ,TvC ,TwC ppp uC ,vC ,wC ppp
22 mW
smJ:unit Latent heat flux, LE
ruL ,rvL ,rwL vvv
2-2
1-
ms
kg
sm
mskg:unit Momentum flux, MO
uu ,uv ,uw
b. Reynolds stressV
X
Z
0wu
wuzx wu vu uu
wv vv uv
ww vw uw
zu
yu
xu
zv
yv
xv
zw
yw
xw
Tensor
Buoyancy flux,
vpvpvp TuC ,TvC ,TwC vpvpvp uC ,vC ,wC
vp TwC
rwT608.0Tw)r608.01(Tw v
7. Mean governing equations in turbulent flow
,T Rp vd
,0z
w
y
v
x
u
vz
vy
vx
z)ww(
y)vw(
x)uw(ˆ
zp̂1
zw
yw
xw
tw
z)wv(
y)vv(
x)uv(
yp̂1
zv
yv
xv
tv
z)wu(
y)vu(
x)uu(
xp̂1
zu
yu
xu
tu
g wvu
uf wvu
vf wvu
00
0
0
tz)w(
y)v(
x)u(
zyxtwvu Q
rz)rw(
y)rv(
x)ru(
zr
yr
xr
tr wvu Q
6. Frictional velocity *U 4/122* )wvwu(U
)wwvvuu(e21
(G) (F) (E) (D) (C) (B) (A)
uuuu i
i
0j
j
0j
i
j xpu1'
x
euiv
gi3x
ujix
ejt
ee
A. Local change term
B. Advection term
C. Shear production term
D. Buoyancy production term
E. Transport term
F. Pressure correlation term
G. Dissipation
8 TKE budget equation
e
zpw1'
zew
vg
zu
te
00
wwu
For horizontal homogeneous condition, x direction along the mean wind direction,mean vertical velocity is zero.
9. Static stability and instability
The atmosphere is unstable if a parcel at equilibrium is displaced slightly upward and finds itself warmer than its environment and therefore continues to rise spontaneously away from its starting equilibrium point.
The atmosphere is stable if a parcel at equilibrium is displacedslightly upward and finds itself colder than its environment andtherefore sink back to its original equilibrium point.
0z
Stable d
0z
Unstable d
10. Thermodynamic structure of atmospheric boundary layer
0z
e
0z
e
s
s
a. Flux Richardson number
zv
zu
v0
g
wvwu
wfR
0R :unstable static f
0R :neutral static f
0R :stable static f
1R :flow) (turbulent unstable dynamic f 1R :flow)turbulent -(non stable dynamic f
b. Gradient Richardson number
11. Richardson number
ci RR
ti RR
0.1R ;25.021.0R tc
Turbulent flow
Non-turbulent flow
])v()u[(
zgB 22
v
vR
c. Bulk Richardson number
2zv2
zu
zv
0v
g
)()(iR
12. Turbulent closure problem
Simplified governing equations
a. First-order closure
Z
13. Monin-Obukhov length L
Static unstable
Static stable
Dynamic unstable
Dynamic stable
Using surface layer relation
Static unstable
Static stable
Dynamic unstable
Dynamic stable
14. Turbulent Analyses
a. Fourier Transform
Why do we need the frequency information?
No frequency information is available in the time-domain signal!
xesin(x)cos(x) ii
...t)sin(b...sin(t)b...t)cos(a...cos(2t)acos(t)aaF(t) 1210
00t)sin(bt)cos(aF(t)
0f
ft2f
0
t ececF(t)
ii
b. Discrete Fourier Transform
Observations: N
Sampling interval: t
t NPeriod
tN1
T1
tkt:nobservatio k at the time th
First harmonic frequency: ... , ..., , ,tN
ntN
2tN
1All frequency:
nth harmonic frequency: tNnf
1N
0n
2Nnk
c(n)eF(k)i
1N
0kNkn2
N
1N
0kNkn2
N1
1N
0k
2N
A(k)
)A(k)sin( )A(k)cos(
ec(n) Nnk
i
i
c. Aliasing, Nyquist frequency, and folding
If sampling rate is , the highest wave frequency can be resolved is ,which is called
sf 2fs
Nyquist frequency
example
If there were a true signal of f=0.9 Hz that was sampled at fs=1.0 Hz, then, onewould find that the signal has been interpreted as the signal of f=0.1 Hz. In otherwords, the real signal f=0.9 Hz was folded into the signal f=0.1Hz.
Folding occurs at Nyquist frequency.
What problem does folding cause?
d. Leakage
e. Detrend, window
f. Energy Spectrum2
imag2
real2 (n)c(n)c|c(n)|
Discrete spectral intensity (or energy) n)(EA
g. Spectral energy density nn)(E
AAn)(S
ff)(E
AAf)(S
h. Turbulent energy cascade Turbulent spectral similarity
•Energy associated with large-scale motion eventually is transferred to the large turbulent eddies.•The large eddies then transport this energy to small-scale eddies.•These smaller scale eddies then transfer the energy to even small-scale eddies..., and so on•Eventually, the energy is dissipated into heat via molecular viscosity.
f2
f2
f2
A
nnfor ,|c(n)|
evenN if ,1n1,....,nfor ,|c(n)|2
oddN if ,n1,....,nfor ,|c(n)|2
n)(E
Inertial sub-range is in an equilibrium state, Kolmogorov assumes that the energy density per unit wave number depends only on the wave number and the rate of energy dissipation. , ),S(
3-2
2-3)E(
1-
sm
sm)S(
m :
I. Kolmogorov's Energy Spectrum
5)(S2
3
3/53/2)S(
wavelength
1 wave-number
speed :M
,Mf
)ln(ln)]ln[S( 3/235
)]ln[S(
)ln(
3
5
3/55/33/2Mf fM)S(
)Mln(ln)]ln[S(f 5/33/235
)]ln[S(f
)ln(f
3
5
16. Ekman Spiral in the atmospheric boundary layer
0)uuf(
0)vvf(
gzv
gzu
2
2
2
2
m
m
k
k
Boundary conditions
z as vv ,uu
0zat 0v 0,u
gg
2/12
fzg )( z),cose1(uu
mk zsineuv z
g
4uv 0,z ,1tan
gg u96.0)e1(uu 0,v zD Ekman layer.
Atmosphere: (m) 1400990 z ,sm 10-5 1-2
mk
Boundary layer vertical secondary circulation
21
y
u
yv
D
0xu gdz)(w(D)
Hurricane 12141310000
20y
usm 5 ,s10f ,s102g
mk
1-2 ms 10w(D)
D
Dynamics of vortex spin down and spin up
convergence
divergence
convection
17. Oceanic Ekman layer
0uf
0vf
2
2
2
2
zv
zu
Boundary condition:
zv
0
zu
0
)wv(
)wu(
0
0
y
x
0;zat
0v
0u;zat
)zcos()zsin([v
)zsin()zcos([u
44fe
44fe
0
z0
z
yx
yxSolution:
2f
4
x
y
V45 ,1tan
M
2 2
,tan
18. Application of Pi theory in the surface
)sm( )''w(
velocityfrictional ,)sm( )wv()wu(/||u
)m( z
32g
222/1222*
ov
ooo
v
sizeeddyorheight
How to represent in terms of relevant parameters:zu
Four variables and two basic units result in two dimensionless numbers, e.g.:
3*
0
* u
z)w(gzu
uz and v
v
The standard way of formulating this is by defining:
0
3*
)w(u
gL
v
v
Monin-Oubkhov length
19. Similarity theory
constant Karman -Von
0.35(0.4),
),()(Lz
zu
uz*
mm
0 51
0)161( 4/1
for
for
m
m
a. Neutral condition 0Lz 1)0( m
1zu
uz*
)ln(u0
*zzu
disappear. windsreheight whe theis z0
b. Non-neutral condition 0Lz
)]()[ln(u0
*
zzu m
0 ,5)(
0,)161( ,tan2)ln()ln(2)( 4/12
12
12
1 2
for
forxx
m
xxm
)()ln(t*
0zz)(
h
vv
Temperature profiles in the surface layer
0 ,5)(
0,)161( ),ln(2)( 2/12
1
for
fory
h
yh
t0 zzat vv
0t zz Normally,
Similarly,
)()ln(q*
0zz
q)qq(
q
)()( hq
q0 zzat qq
0t zz Normally,
20. Bulk transfer relations
).qq(u)qw(
),(u)w(
,uu
00
v0v0v
22*
Q
H
D
C
C
C
: , , QHD CCC Drag coefficient of momentum, heat, and moisture.
,)( 20
2*
)]()z/z[ln(
2uu
mDC
,)( 20
2*
)]z/z[ln(
2uu DNC
,)]()z/z)][ln(()z/z[ln( t0
2
hmHC
,)]z/z)][ln(z/z[ln( t0
2HNC
,)]()z/z)][ln(()z/z[ln( q0
2
qmQC
,)]z/z)][ln(z/z[ln( q0
2QNC
20. The surface energy balance
storageenergy :G ,FSHLERG eos
)K m (Jcapacity heat :C ,CG 1-2-gt
Tg
g
Difference between heat capacity and specific heat.
19. Flux footprint
Flux footprint describes a dependence of vertical turbulent fluxes, such as, heat, water, gas, and momentum transport, on the condition of upwind area seen by the Instruments. Another frequently used term representing the same concept is fetch.
atmosphere
Land orOcean
sR
LE SH
eoF
1-3-1-3-p K m JK kg Jm kg :C
Diurnal variation of surface energy budget over land
sRLE
SH
eoF
sR
SH
LE
eoF
oB
Wet surface
Dry surface
Radiative heating at the surface
lwlwswsws FFFFR )1(FFF sswswsw
21. Convective Boundary Layer
Turbulent
Potential temperature (K) Buoyancy fluxes (K m/s)
Mixed layer model
turbulence
CBL Growth
d
v
q
q
h
subsidence
Entrainment drying
Entrainment warming
0vw 0qw
Mixed Layer
1. ML warming caused by heat input from the surface and entrainment
2. Growth of the CBL controlled by entrainment and subsidence
3. ML moistening or drying due to surface evaporation and entrainment
hvw hqw
Empirical relations in the mixed layer
2zz3/2
zz
ww )8.01()(8.1
ii2*
2 )1.11()(8.0
ii3*
3
zz
zz
ww
Some important relations under the mixed layer model framework
hvhz
0vhz
zv )w()w()1()w(
0v )w(
hv )w( h
25.00v
hv
)w()w(
Deardorff convective velocity scale *w
3/10vi
g* ])w(z[w
v
:zi :)w( 0vMixed layer depth Surface buoyancy flux
3/1z
zv
g* ]dzw5.2[w
i
0v or
Narrow branch of updraft compensated by broad branch of downdraft
22. Convective plume structures, skewness, and Kurtosis
Skewed distribution
Skewness
2/32
3
)w(w
Kurtosis
322
4
)w(w
23. Nocturnal boundary layer
Nocturnal jet:
Nocturnal jet forms at night-time overland under clear sky conditions. The wind speed may be significantly super-geostrophic.
Inertial oscillation theory
zwv
gtv
zwu
tu
fufu
fv
Governing equations
Further assuming daytime boundary layer is in a steady state
0 ;0t
vt
u dd
uddzwu
f1
d
vdgdzwv
f1
gd
F)(v
Fu)(uu
After sunset, nocturnal boundary layer forms, the air above the NBL can be assumed to be free atmosphere, the governing equation becomes
ngtv
ntu
fufu
fv
n
n
)u(uf gn2
t
u2n
2
It has a solution in the format of
Bcos(ft)sin(ft)A uu gn
Bsin(ft)Acos(ft)vn
Initial condition 0 t ;v v;uu dndn
cos(ft)Fsin(ft) Fuu vdudgn
sin(ft)Fcos(ft) Fv vdudn f2T
Solution
gu
10
2 3
6
45
7
8
9101112
13
14
15
Influence of slope z
gi
ge
zwv
gtv
zwu
gtu
fufu
sin gfvfvve
v
24. Inflection-point instability in rotation-shear flow
uv
Vorticity maximumInflection point
Barotropic Ekman flow with constant Km (the simplest PBL flow)
ξ×1000 x
y
zξ Rol
l axisVg
ε
z)cose1(uu zg
zsineuv zg
2/12
f )(mk
In the roll-coordinate, the vorticity equation of horizontal homogeneous Boussinesq flow
frictioncosv0
*
yg
yu
zu
tff
axis- xandeast between angle :
cos2;sin2;)( *v
ffz
zyw
Procedure for solving the problem (classic linear method)
1. Using small perturbation method to linearize equation
2. Assuming simple harmonic wave solution
)(e ˆ ctyim m is the wavenumber; c is the complex eigenvalue withreal part the wave velocity and imaginary part the growth rate.
3. Obtaining Rayleigh necessary condition for instability
Wavenumber m
The maximum growth rate of 0.028 occurs at wavenumber 0.5 and oriented 18o to the left of the geostrophic wind.(Brown 1972 JAS)
Intertropiccal Convergence Zone (ITCZ)
Trade cumulus
Transition
Stratus and stratocumulus
subsidence
Trade wind inversion
St & Sc
St &
Sc
25. Boundary layer clouds
Cloud radiative effects depend on cloud distribution, height, and optical properties.
gT
cT
cg TT
cT aT
ac TT
Low cloud High cloud
SW cloud forcing dominates LW cloud forcing dominates
SW cloud forcing = clear-sky SW radiation – full-sky SW radiation
LW cloud forcing = clear-sky LW radiation – full-sky LW radiation
Net cloud forcing (CRF) = SW cloud forcing + LW cloud forcing
In GCMs, clouds are not resolved and have to be parameterized empirically in terms of resolved variables.
water vapor (WV) cloud surface albedo lapse rate (LR) WV+LR ALL
Aerosol feedback
Direct aerosol effect: scattering, reflecting, and absorbing solar radiation by particles.
Primary indirect aerosol effect (Primary Twomey effect): cloud reflectivity is enhanced due to the increased concentrations of cloud droplets caused by anthropogenic cloud condensation nuclei (CNN).
Secondary indirect aerosol effect (Second Twomey effect):
1. Greater concentrations of smaller droplets in polluted clouds reduce cloud precipitation efficiency by restricting coalescence and result in increased cloud cover, thicknesses, and lifetime.
Mechanisms of maintaining cloud-topped boundary layer
1. Surface forcing
2. Cloud top radiative cooling
3. Cloud top evaporative cooling