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