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