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

Ug

Ekman Layer Height

mf

KhE 300~

2

Sutton (1953) used this as the ABL height under

neutral conditions

Tennekes (1982)

EhuK *

kmf

uhABL 1~

25.0 *

With Stratification? -

- Stable or Unstable

Lecture 2a

Convective Boundary Layers

And

Convective Flows

ABL EVOLUTION

287 288 289 290 291 292 293 294

Virtual potential temperature (K)

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

z(m

)

287 288 289 290 291 292 293 294

Virtual potential temperature (K)

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

287 288 289 290 291 292 293 294

Virtual potential temperature (K)287 288 289 290 291 292 293 294

Virtual potential temperature (K)

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

z(m

)

0T

tqhN 022 ≈

2

1

212

1

2

02 t

N

qh

Non-

Penetrative

and

Penetrative

Convection

fluxbuoyancy

C

gQq

p00

=

ABL EVOLUTION

287 288 289 290 291 292 293 294

Virtual potential temperature (K)

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

z(m

)

287 288 289 290 291 292 293 294

Virtual potential temperature (K)

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

287 288 289 290 291 292 293 294

Virtual potential temperature (K)287 288 289 290 291 292 293 294

Virtual potential temperature (K)

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

0

20

40

60

80

100

120

140

160

180

200

220

1641-1644

1657-1701

1721-1724

1738-1741

1756-1800

1814-1817

1835-1838

1856-1900

z(m

)

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

Brazel et al. 2000

March 19, 2008, 5pm to 10pm

Infrared imaging of Phoenix

UHI Experiment

Variation or horizontal variance

(normalized). (Solid Curve --

Laboratory)

(Fernando et al., Dyn. Atmos.

Oceans, 13,95-121, 1989)

Variation of vertical variance

(normalized).

Convective Scaling Vs. Data

Wind Shear Found at all

Altitudes