Arya S.P.S. Introduction to Micrometeorology

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218 13 Marine Atmospheric Boundary Layer

measured wind profiles during the same experiment showed a wide range

of shapes and forms and

are

quite sensitive to stability, baroclinity, and

presence

clouds in the

boundary

layer. In short, there is no typical

wind profile for the MABL. Figure 6.6 is an example of wind component

profiles in a trade-wind boundary layer.

8 0

~OO

Db~

O"u,v,w

@ . 0

~8°

Qlo

f15'

0l>.M"l>.

l>.

It>

~l>.'7:,.

l>. l>.

l>.~l>.

l>. l>.f;t,.

l>.

l>.l>.

l>.

iii.

0.1

0.2

0.5 I 2

5 10

-ZiL

0"8

- 8*

3 0

§o

o 0

0.1

0.2

0.5

I 2

5 10

-ZIL

Fig. 13.13 Normalized

standard

deviations of velocity and temperature fluctuations in

the marine surface

layer

as functions of stability during BOMEX. D, a-ju*;

oJu i:

to,

uju*.

[After

Leavitt

and Paulson (1975).]

13.5 Turbulence over

Water

219

13.5

TURBULENCE

OVER

WATER

There is some observational evidence indicating that very close to the

wavy

sea

surface, turbulence in air is modulated and affected by surface

waves (see, e.g., Kitaigorodski, 1970). A few wave heights above the

mean

sea

level, however, the direct influence of waves becomes insignifi-

cant and the fully developed surface layer turbulence is similar to that

over

homogeneous land surfaces. This means that the appropriate scales

velocity, temperature,

and

humidity fluctuations are ll*,

(J*,

and

q*,

while the ratio of the height scales zlL is the appropriate stability or

similarity parameter.

The normalized standard deviations of velocity fluctuations in the sur-

face layer, measured during the 1969 Barbados Oceanographic anu Mete-

orological Experiment (BOMEX), are plotted against

-ziL

in Fig. 13.13.

Note

that while

(J"

u; apparently approaches the local free convection

prediction

(J"wlll*

(-zIL)I/3, for large values of

-zIL,

aju*

and

a.lu

;

not

show

any

such

behavior. Other data

over

land and ocean surfaces

also indicate

that

horizontal components of turbulence do not follow ei-

6...--------,r-----,------,-----=-71

f::,.

400

300

100

2O::::"""

~'--

--J.

-J..

-J

200

h/L

Fig. 13.14 Normalized

standard

deviations of horizontal velocity fluctuations in the

surface

layer

as functions of hlL

compared

with mixed-layer similarity relations. ---, (Jlu. =

[4 + 0.6( -hIL)1J3]1/2;

--,

alu

; = (12 - 0.5 hIL)Jl3;

1'1,

airplane

data;

Minnesota

tower

data. [After

Panofsky

perrnission.]

220 13 Marine Atmospheric Boundary Layer

7654

- z

----L

...J-

----,JL-

-'--

---'-

--'-----'

Fig.13.15 Normalized standard deviation of vertical velocity fluctuations in the surface

layer as a function

zlL

compared

with surface layer similarity relations. ---,

a.Ju

; =

[1.6 + 2.9(-zIL)213]1!2;

--,

a.Lu

; = 1.3[1 - 3(zIL)]1I3;

airplane data;

Minnesota

tower

Reprinted by permission.]

ther

Monin-Obukhov

or local free convection similarity scaling, but they

seem to follow the mixed-layer scaling, even in the surface layer, i.e., for

-h/L

p I, o ;

~W*

and (Tv

W*. This implies that

(Tuju*

(-h/L)II3.

Figures 13.14 and 13.15 show different behaviors of horizontal and verti-

cal components of turbulence and suggest different empirical relations for

them in unstable

and

convective conditions.

13.6

APPLICATIONS

The material presented in this chapter may have applications

III

the

following areas:

• Determining energy budgets of an ocean or

sea

surface

• Determining the

air-sea

exchanges of momentum, heat, and water

vapor at the sea surface

• Predicting

sea

state, including storm surges

• Modeling atmospheric and oceanic boundary layers and the interac-

tion

between

the two

Problems and Exercises

221

• Modeling

upper

oceanic currents and circulations

• Forecasting storm tracks and weather at

sea

• Long-range

weather

forecasting and climate simulations, using cou-

pled

upper-ocean-atmospheric

models

PROBLEMS

AND

EXERCISES

1. The following observations were made during a period of intense cold

air

outbreak

over

the

East

China

Sea

during the 1974Air Mass Trans-

formation Experiment:

Sensible heat flux from the sea surface

= 220 W m?

Latent

heat flux from the

sea

surface = 490 W rn?

Net

radiative loss from the sea surface = 30 W m?

Sea-surface temperature = 21.5°C

Oceanic mixed-layer depth

= 100 m

Calculate the

rate

of warming or cooling of the oceanic mixed layer in

°C

per

day.

2. Calculate

and

compare the wavelengths and wave speeds of sea-sur-

face swells of periods 10, 20, 30, 40, 50, and 60 sec.

3. Using

Charnock's

relation for the surface roughness and the logarith-

mic wind-profile law, derive Eq. (13.5) and plot C

as a function of

the

Froude

number

F = UI(gZ)l/2 in the wind speed range of 5 to 50 m

sec-

at z = 10 m.

4. (a) Starting from Eq. (11.16), derive Eq. (13.9).

(b) Calculate and plot

CD/C

as a function of

RiB

in the range

-0.1

RiB

0.1, for C

= 1.5 X 10-

5. The following measurements were made from the research vessel

FLIP

during the 1969 Barbados Oceanographic and Meteorological

Experiment (BOMEX):

Mean wind speed at a

1O.9-m

height = 7.96 m sec"

Mean temperature at a

1O.9-m

height = 27.34°C

Mean specific humidity at a

1O.9-m

height = 16.04 g kg-

Sea-surface temperature = 28.35°C

Surface

pressure

= 1000 mbar

Using the bulk transfer formulas with

= C

= 1.5 X 10-

calculate the surface stress, the sensible heat flux, the rate of evapora-

tion,

and

the Bowen ratio at the time of these observations.

6. Mean profile measurements made during BOMEX at the same time as

222 13 Marine Atmospheric Boundary Layer

Problem

are

given as follows:

Height Wind

speed

Temperature

Specific-humidity

(m)

sec

)

eC)

(g kg-

)

2.42

7.33 27.55

16.72

3.73

7.58 27.50

16.49

6.18 7.80 27.45

16.29

10.90 7.96 27.34

16.04

(a)

Use

the

profile

method

with

the

appropriate

Moniu-Obukhov

flux-profile relations to calculate us ,

()*,

and q«.

(b)

Compare

the

fluxes obtained from the profile method with those

from the bulk transfer approach.

Chapter 14

Nonhomogeneous Boundary

Layers

14.1 TYPES OF SURFACE INHOMOGENEITIES

Micrometeorological theories, observations, and methods discussed in

the preceding chapters are strictly applicable to the atmospheric bound-

ary

layer

flow

over

a flat, uniform, and homogeneous terrain. Over such

an ideal surface, the

PBL

is also horizontally homogeneous and in equilib-

rium with the local surface characteristics, especially

when

synoptic con-

ditions do not change rapidly. Such a simple state of affairs may exist

frequently

over

open

oceans, seas, and large lakes, as well as over exten-

sive desert, ice, snow, prairie, and forest areas of the world. More often,

though, land surfaces are characterized by surface inhomogeneities,

which make the atmospheric boundary layers over them also nonhomoge-

neous.

Surface inhomogeneities, which have important effects on atmospheric

flows, include boundaries between land and water surfaces (coastlines);

the transitions between urban and rural areas or between different types

of vegetation; mesoscale oceanic eddies, surface currents, and other re-

gions of varying sea-surface temperature; and hills and valleys. In going

over

these terrain inhomogeneities, the flow encounters sudden or gradual

changes in surface roughness, temperature, wetness, or elevation. Quite

often, changes in several surface characteristics

occur

together, and a

complete understanding of modifications in

PBL

properties (velocity,

temperature, and humidity) requires that changes in the surface rough-

ness, temperature, wetness, and elevation be treated simultaneously.

For

the sake of simplicity and convenience, however, micrometeorologists

have studied the effects of changes in the surface roughness, temperature,

etc., separately. Due to nonlinearity of the system, however, the individ-

ual effects may not simply be added or superimposed; the overall effect of

a combination of changes in surface characteristics may differ substan-

tially from the sum of individual effects. There are only a few studies of

223

224 14 Nonhomogeneous Boundary Layers

the combined effects of two or more types of surface inhomogeneities

occurring simultaneously.

14.2

STEP

CHANGES

SURFACE

ROUGHNESS

Modification of boundary layer following a step change in the surface

roughness normal to the direction of flow has been extensively studied

experimentally as well as theoretically. A schematic of the approach

flow

and the modified flow due to change in the surface roughness is shown in

Fig. 14.1 for the case of neutral stability. Note that the approach wind-

profile

U1(z) is a function of the upwind friction velocity

U*l

and the

surface roughness

ZOI • Following the roughness change from ZOl to

Z02,

the

friction velocity is modified and so is the wind profile near the new sur-

face. Modifications to mean wind profile and turbulence as indicated by

computed variances

and

fluxes are found to be confined to a layer whose

thickness increases with distance from the line of discontinuity in the

surface.

The

modified layer is commonly referred to as an internal bound-

ary layer (IBL), because it grows within another boundary layer associ-

ated with the

approach

flow.

PBL

TOP

UI(z)

2(z,x)

IBL

hj(x)

zOI

z02

Fig. 14.1

Schematic

of the internal

boundary

layer development and wind profile modifi-

cation following a step change in the surface roughness.

14.2 Step Changes in Surface Roughness 225

Above

the

IBL,

flow characteristics are the same as in the approach

flow at

the

same

height

above

the surface.

These

are essentially deter-

mined by upwind surface characteristics.

The

influence

the upstream

surface

may

expected

disappear

at a sufficiently large distance from

the roughness discontinuity,

where

the

IBL

has grown to the equilibrium

PBL

depth

for

the

new

surface, or it is otherwise limited by a strong low-

level inversion.

some

the

earliest experimental studies of flow

over

step

change

in surface roughness,

hundreds

of bushel

baskets

or Christmas trees were

placed on a

frozen

lake

(Lake

Mendota,

Wisconsin) in winter and modifi-

cations to

the

near-surface wind and temperature profiles due to the

change in roughness

were

studied (see,

e.g.,

Kutzback, 1961; Stearns and

Lettau,

1963).

The

observed

wind profiles did not

extend

beyond a height

of 3.2 m

and

independent

shear

stress

or drag measurements were

made in

these

experiments.

comprehensive

field study was

conducted

by Bradley (1968),

who

measured

changes in

the

surface

shear

stress and wind profiles with

distance from

the

roughness discontinuity in going from a comparatively

"smooth"

(but aerodynamically rough)

tarmac

surface

(zo

= 0.02 mm) to

moderately

rough surface

wire

mesh

with spikes (zo = 2.5 mm), and

vice versa. Figure 14.2 shows the variations in surface stress with dis-

tance

for

the

two

cases

(smooth to rough and rough to smooth) of the

internal

boundary

layer

flow

under

near-neutral stability conditions. Here

the local

stress

is normalized by the magnitude

the equilibrium stress

for the upwind surface.

Note

that

the

observations

Fig. 14.2 indicate that there is a

rather

sharp

change

the

surface stress

near

the roughness discontinuity, with a

considerable

amount

overshoot

above

undershoot

below its ex-

pected

equilibrium value for

the

new (rougher or smoother) surface. Re-

laxation to the equilibrium value

occurs

very rapidly in the first few me-

ters

and

gradually thereafter. No detectable changes in the surface stress

are

apparent

from

observations

beyond

a distance

10-15 m from the

roughness discontinuity.

But

number

of theoretical and numerical

model studies

predict

slow adjustment occurring

over

much longer dis-

tances (see,

e.g.,

Elliot, 1958; Panofsky

and

Townsend,

1964; Rao et al.,

1974); some

these

theoretical results are shown in Fig. 14.2.

Observed

changes in

the

wind profile, following

step

changes in the

surface roughness from

smooth

to rough

and

vice versa, are shown in Fig.

14.3. At

any

downwind

distance from

the

surface discontinuity, the lower

part

the

wind profile is representative

the local surface, while the

upper

part

represents

the

upwind surface.

Within

the

lowest

10-15%

the

PBL

the

velocity profile in neutral

226 14 Nonhomogeneous Boundary Layers

o 4

...

... 2

(a)

•

t.;~~-P-T-

T X

P-T

100m-

2 4

8 10

(m)

12 14

(b)

0.3

x 00

x ox

°ox

0.1

(m)

Fig. 14.2 Observed variations of surface shear stress downwind of (a) smooth to rough

(from

Zo = 0.02 mm to Zo = 2.5 mm) and (b) rough to smooth (from Zo = 2.5 mm to Zo = 0.02

mm) transitions compared with certain theories.

"',

Elliot (1958);

--,

Panofsky and Town-

send (1964). [After Bradley (1968).]

stability

may

be approximated as (Elliot, 1958)

U = (U*2/k) In(z/

Z02),

U = (U*I/k) In(z!ZOI),

for z

o:s;

for z > hi

(14.1)

In reality, the

IBL

not

as sharply defined as is implied by the discontinu-

ity (kink) in the velocity profile [Eq. (14.1)], and there is a finite blending

region around

z = hi in which the velocity profile gradually changes from

one

form to another.

There

is also a question about the validity of the log

law in the

lower

layer, especially in the region where the friction velocity

might be changing rapidly and has not reached its equilibrium value

U*2.

The more appropriate equations

mean motion in the lower part of the

14.2

Step

Changes in Surface Roughness 227

(a )

1.00

0.10

xlm)

+ 16-42

6"42

DATA 0 2·32

x 1·18

•

0·32

1.1

1.0

0.9

0.60.5

0.01

L..-_--'-

....L..

.L-_........L

--I-

...L-_--J

0.4

0.7 0.8

U/U

Fig. 14.3

Observed

modifications of velocity profiles downwind of (a) smooth to rough

(from

= 0.02 mm to Zo = 2.5 mm) and (b) rough to smooth (from

= 2.5 mm to z" = 0.02

mm) transitions

compared

with Panofsky and

Townsend's

(1964) theory. [After Bradley

(1968).]

IBL,

where the Coriolis effects can be neglected, are

u(au/ax)

+ W(aUlaz) =

aTzx/aZ

au/ax

+ aw/az = 0

(14.2)

which form the basis for simpler theoretical models.

Note

that

eliminating

the

pressure

gradient and Coriolis acceleration terms from Eq. (14.2)

would limit their validity to

rather

short distances from the surface dis-

continuity.