ji nie's group · january 27, 2004 9:4 elsevier/aid aid 5.3 planetary boundary layer momentum...

44

Upload: others

Post on 05-Oct-2020

2 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

����

�������

Page 2: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

5

31

6.

2

2

4

Page 3: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

3 %%

23 1 %- %

0 %

Page 4: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

January 27, 2004 9:4 Elsevier/AID aid

5.3 planetary boundary layer momentum equations 123

−f!u − ug

"− ∂v′w′

∂z= 0 (5.19)

where (2.23) is used to express the pressure gradient force in terms of geostrophicvelocity.

5.3.1 Well-Mixed Boundary Layer

If a convective boundary layer is topped by a stable layer, turbulent mixing canlead to formation of a well-mixed layer. Such boundary layers occur commonlyover land during the day when surface heating is strong and over oceans when theair near the sea surface is colder than the surface water temperature. The tropicaloceans typically have boundary layers of this type.

In a well-mixed boundary layer, the wind speed and potential temperature arenearly independent of height, as shown schematically in Fig. 5.2, and to a firstapproximation it is possible to treat the layer as a slab in which the velocity andpotential temperature profiles are constant with height and turbulent fluxes varylinearly with height. For simplicity, we assume that the turbulence vanishes at thetop of the boundary layer. Observations indicate that the surface momentum fluxcan be represented by a bulk aerodynamic formula3

#u′w′

$

s= −Cd

%%V%% u, and

#v′w′

$

s= −Cd

%%V%% v

Fig. 5.2 Mean potential temperature, θ0, and mean zonal wind, U , profiles in a well-mixed boundarylayer. Adapted from Stull (1988).

3 The turbulent momentum flux is often represented in terms of an “eddy stress” by defining, forexample, τex = ρou′w′. We prefer to avoid this terminology to eliminate possible confusion withmolecular friction.

�����

Page 5: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

��

Page 6: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

/

Page 7: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

/ 5 . /

:

Page 8: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

: ) (- 5

1 :02 :0

第五章:量纲分析

聂绩

1 相似原理

下标 t表示实物,下标 m表示模型。

δl =ltlm

δv =vtvm

δt =tttm

δt =δlδv

δF = FtFm

δF = FtFm

= ρtτtatρmτmam

= δρδ3lδlδ2t

= δρδ2l δ2v

2 无量纲化

∂v∂t + v ·∇v = −1

ρ∇p+ gk + µρ∇

2v + F

LV T

∂v′∂t′ + v′ ·∇v′ = − P

ΠV 21ρ′∇p′ + L

V 2gk + µΠV L

1ρ∇

2v′ +!F

ΠV 2L2F ′

1

第五章:量纲分析

聂绩

1 相似原理

下标 t表示实物,下标 m表示模型。

δl =ltlm

δv =vtvm

δt =tttm

δt =δlδv

δF = FtFm

δF = FtFm

= ρtτtatρmτmam

= δρδ3lδlδ2t

= δρδ2l δ2v

2 无量纲化

∂v∂t + v ·∇v = −1

ρ∇p+ gk + µρ∇

2v + F

LV T

∂v′∂t′ + v′ ·∇v′ = − P

ΠV 21ρ′∇p′ + L

V 2gk + µΠV L

1ρ∇

2v′ +!F

ΠV 2L2F ′

1

2 ⽆量纲化 2

Sr = LV T

Eu = PΠV 2

Fr =!

V 2

gL

Re = ΠV Lµ

Ne ="F

ΠV 2L2

SrtSrm

=( LV T )t

( LV T )m

= δlδvδt

= 1

EutEum

= δpδρδ2v

= 1

FrtFrm

= δ2vδlδg

= 1

RetRem

= δµδρδvδl

= 1

NetNem

= δFδρδ2vδ

2l= 1

Page 9: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23
Page 10: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

�������

第⼋章:湍流

聂绩

A = A+ A′

1

������� �������

Page 11: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

sA

sA

BABAAB

BABA

A

AA

¶¶

=¶¶

¢¢+=

+=+

=

0

,

Page 12: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

rr =

( )TRTRTRp

TTRpp

rrr

r

=¢+=

¢+=¢+ ,

���

� ��

! = #$%

����

Page 13: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

( ) ( )[ ] ( )[ ] ( )[ ] 0=¶

¢+¶+

¶¢+¶

¢+¶+

¶¢+¶

zww

yvv

xuu

trrrrr

0)()()(=

¶¶

¶+

¶¶

+¶¶

zw

yv

xu

trrrr

0=¶

¢¶+

¶¢¶

¢¶zw

yv

xu )()()( rrr

������

Page 14: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

�����

( ) ( ) úû

ùêë

é ¢¢-¶¶

+¢¢-¶¶

+¢¢-¶¶

++¶¶

-=- )(11 wuz

vuy

uux

Fxpvf

dtud

x rrrrr

( ) ( ) úû

ùêë

é ¢¢-¶¶

+¢¢-¶¶

+¢¢-¶¶

++¶¶

-=+ )(11 wvz

vvy

uvx

Fypuf

dtvd

y rrrrr

( ) ( ) úû

ùêë

é ¢¢-¶¶

+¢¢-¶¶

+¢¢-¶¶

++-¶¶

-= )(11 wwz

vwy

uwx

Fgzp

dtwd

x rrrrr

Page 15: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

( ) ( ) ÷÷ø

öççè

æ ¢¢¶¶

+¢¢¶¶

+¢¢¶¶

+¶¶

+¶¶

+¶¶

=

÷÷ø

öççè

涢¶¢+

¶¢¶¢+

¶¢¶¢+

¶¶

+¶¶

+¶¶

=

¶¢¶¢+

¶¢¶¢+

¶¢¶¢+

¶¶

+¶¶

+¶¶

=

¶¶

+¶¶

+¶¶

)(1

1

wuz

vuy

uuxz

uwyuv

xuu

zuw

yuv

xuu

zuw

yuv

xuu

zuw

yuv

xuu

zuw

yuv

xuu

zuw

yuv

xuu

rrrr

rrrr

� ��������������������

Page 16: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

j

jii xf

¶¶

=t

r1

÷÷÷

ø

ö

ççç

è

æ

¢¢-¢¢-¢¢-

¢¢-¢¢-¢¢-

¢¢¢¢-¢¢-=

÷÷÷

ø

ö

ççç

è

æ=

wwvwuwwvvvuvwuvuuu

zzyzxz

zyyyxy

zxyxxx

rrrrrrrrr

ttttttttt

t

���� �

2(� �0�

2(� !���'#���"�)!�3�!� ����!�0� �"�.�!

&�/���&�0!��*,���1%+�3$!�0�-0�

wuzx ¢¢-= rt

Page 17: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

��������������� ���

�������

January 27, 2004 9:4 Elsevier/AID aid

5.1 atmospheric turbulence 119

Noting that the mean velocity fields satisfy the continuity equation (5.5), we canrewrite (5.7), as

Du

Dt= Du

Dt+ ∂

∂x

!u′u′

"+ ∂

∂y

!u′v′

"+ ∂

∂z

!u′w′

"(5.8)

whereD

Dt= ∂

∂t+ u

∂x+ v

∂y+ w

∂z

is the rate of change following the mean motion.The mean equations thus have the form

Du

Dt= − 1

ρ0

∂p

∂x+ f v −

#∂u′u′

∂x+ ∂u′v′

∂y+ ∂u′w′

∂z

$

+ Frx (5.9)

Dv

Dt= − 1

ρ0

∂p

∂y− f u −

#∂u′v′

∂x+ ∂v′v′

∂y+ ∂v′w′

∂z

$

+ Fry (5.10)

Dw

Dt= − 1

ρ0

∂p

∂z+ g

θ

θ0−

#∂u′w′

∂x+ ∂v′w′

∂y+ ∂w′w′

∂z

$

+ Frz (5.11)

Dt= − w

dθ0

dz−

#∂u′θ ′

∂x+ ∂v′θ ′

∂y+ ∂w′θ ′

∂z

$

(5.12)

∂u

∂x+ ∂ v

∂y+ ∂w

∂z= 0 (5.13)

The various covariance terms in square brackets in (5.9)–(5.12) represent turbu-lent fluxes. For example, w′θ ′ is a vertical turbulent heat flux in kinematic form.Similarly w′u′ = u′w′ is a vertical turbulent flux of zonal momentum. For manyboundary layers the magnitudes of the turbulent flux divergence terms are of thesame order as the other terms in (5.9)–(5.12). In such cases, it is not possible toneglect the turbulent flux terms even when only the mean flow is of direct interest.Outside the boundary layer the turbulent fluxes are often sufficiently weak so thatthe terms in square brackets in (5.9)–(5.12) can be neglected in the analysis oflarge-scale flows. This assumption was implicitly made in Chapters 3 and 4.

The complete equations for the mean flow (5.9)–(5.13), unlike the equations forthe total flow (5.1)–(5.5), and the approximate equations of Chapters 3 and 4, arenot a closed set, as in addition to the five unknown mean variables u, v, w, θ , p,

there are unknown turbulent fluxes. To solve these equations, closure assump-tions must be made to approximate the unknown fluxes in terms of the fiveknown mean state variables. Away from regions with horizontal inhomogeneities

January 27, 2004 16:17 Elsevier/AID aid

2.7 thermodynamics of the dry atmosphere 51

Taking the logarithm of (2.44) and differentiating, we find that

cpD ln θ

Dt= cp

D ln T

Dt− R

D ln p

Dt(2.45)

Comparing (2.43) and (2.45), we obtain

cpD ln θ

Dt= J

T= Ds

Dt(2.46)

Thus, for reversible processes, fractional potential temperature changes are indeedproportional to entropy changes. A parcel that conserves entropy following themotion must move along an isentropic (constant θ ) surface.

2.7.2 The Adiabatic Lapse Rate

A relationship between the lapse rate of temperature (i.e., the rate of decrease oftemperature with respect to height) and the rate of change of potential tempera-ture with respect to height can be obtained by taking the logarithm of (2.44) anddifferentiating with respect to height. Using the hydrostatic equation and the idealgas law to simplify the result gives

T

θ

∂θ

∂z= ∂T

∂z+ g

cp(2.47)

For an atmosphere in which the potential temperature is constant with respect toheight, the lapse rate is thus

−dT

dz= g

cp≡ #d (2.48)

Hence, the dry adiabatic lapse rate is approximately constant throughout the loweratmosphere.

2.7.3 Static Stability

If potential temperature is a function of height, the atmospheric lapse rate, # ≡−∂T/∂z, will differ from the adiabatic lapse rate and

T

θ

∂θ

∂z= #d − # (2.49)

If # < #d so that θ increases with height, an air parcel that undergoes an adiabaticdisplacement from its equilibrium level will be positively buoyant when displaced

Page 18: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

January 27, 2004 9:4 Elsevier/AID aid

5.1 atmospheric turbulence 119

Noting that the mean velocity fields satisfy the continuity equation (5.5), we canrewrite (5.7), as

Du

Dt= Du

Dt+ ∂

∂x

!u′u′

"+ ∂

∂y

!u′v′

"+ ∂

∂z

!u′w′

"(5.8)

whereD

Dt= ∂

∂t+ u

∂x+ v

∂y+ w

∂z

is the rate of change following the mean motion.The mean equations thus have the form

Du

Dt= − 1

ρ0

∂p

∂x+ f v −

#∂u′u′

∂x+ ∂u′v′

∂y+ ∂u′w′

∂z

$

+ Frx (5.9)

Dv

Dt= − 1

ρ0

∂p

∂y− f u −

#∂u′v′

∂x+ ∂v′v′

∂y+ ∂v′w′

∂z

$

+ Fry (5.10)

Dw

Dt= − 1

ρ0

∂p

∂z+ g

θ

θ0−

#∂u′w′

∂x+ ∂v′w′

∂y+ ∂w′w′

∂z

$

+ Frz (5.11)

Dt= − w

dθ0

dz−

#∂u′θ ′

∂x+ ∂v′θ ′

∂y+ ∂w′θ ′

∂z

$

(5.12)

∂u

∂x+ ∂ v

∂y+ ∂w

∂z= 0 (5.13)

The various covariance terms in square brackets in (5.9)–(5.12) represent turbu-lent fluxes. For example, w′θ ′ is a vertical turbulent heat flux in kinematic form.Similarly w′u′ = u′w′ is a vertical turbulent flux of zonal momentum. For manyboundary layers the magnitudes of the turbulent flux divergence terms are of thesame order as the other terms in (5.9)–(5.12). In such cases, it is not possible toneglect the turbulent flux terms even when only the mean flow is of direct interest.Outside the boundary layer the turbulent fluxes are often sufficiently weak so thatthe terms in square brackets in (5.9)–(5.12) can be neglected in the analysis oflarge-scale flows. This assumption was implicitly made in Chapters 3 and 4.

The complete equations for the mean flow (5.9)–(5.13), unlike the equations forthe total flow (5.1)–(5.5), and the approximate equations of Chapters 3 and 4, arenot a closed set, as in addition to the five unknown mean variables u, v, w, θ , p,

there are unknown turbulent fluxes. To solve these equations, closure assump-tions must be made to approximate the unknown fluxes in terms of the fiveknown mean state variables. Away from regions with horizontal inhomogeneities

��������

Page 19: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

��� �����

Page 20: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

���!������ ���� ����������������

� ����������

( )i ji

j

u uf U

Page 21: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

-

January 27, 2004 9:4 Elsevier/AID aid

5.3 planetary boundary layer momentum equations 125

Fig. 5.3 Balance of forces in the well-mixed planetary boundary layer: P designates the pressuregradient force, Co the Coriolis force, and FT the turbulent drag.

means is needed to determine the vertical dependence of the turbulent momentumflux divergence in terms of mean variables in order to obtain closed equations forthe boundary layer variables. The traditional approach to this closure problem isto assume that turbulent eddies act in a manner analogous to molecular diffusionso that the flux of a given field is proportional to the local gradient of the mean. Inthis case the turbulent flux terms in (5.18) and (5.19) are written as

u′w′ = − Km

!∂u

∂z

"; v′w′ = − Km

!∂ v

∂z

"

and the potential temperature flux can be written as

θ ′w′ = − Kh

!∂θ

∂z

"

where Km(m2s− 1) is the eddy viscosity coefficient and Khis the eddy diffusivityof heat. This closure scheme is often referred to as K theory.

The K theory has many limitations. Unlike the molecular viscosity coefficient,eddy viscosities depend on the flow rather than the physical properties of thefluid and must be determined empirically for each situation. The simplest modelshave assumed that the eddy exchange coefficient is constant throughout the flow.This approximation may be adequate for estimating the small-scale diffusion ofpassive tracers in the free atmosphere. However, it is a very poor approximationin the boundary layer where the scales and intensities of typical turbulent eddiesare strongly dependent on the distance to the surface as well as the static stability.Furthermore, in many cases the most energetic eddies have dimensions comparableto the boundary layer depth, and neither the momentum flux nor the heat flux isproportional to the local gradient of the mean. For example, in much of the mixedlayer, heat fluxes are positive even though the mean stratification may be very closeto neutral.

January 27, 2004 9:4 Elsevier/AID aid

5.3 planetary boundary layer momentum equations 125

Fig. 5.3 Balance of forces in the well-mixed planetary boundary layer: P designates the pressuregradient force, Co the Coriolis force, and FT the turbulent drag.

means is needed to determine the vertical dependence of the turbulent momentumflux divergence in terms of mean variables in order to obtain closed equations forthe boundary layer variables. The traditional approach to this closure problem isto assume that turbulent eddies act in a manner analogous to molecular diffusionso that the flux of a given field is proportional to the local gradient of the mean. Inthis case the turbulent flux terms in (5.18) and (5.19) are written as

u′w′ = − Km

!∂u

∂z

"; v′w′ = − Km

!∂ v

∂z

"

and the potential temperature flux can be written as

θ ′w′ = − Kh

!∂θ

∂z

"

where Km(m2s− 1) is the eddy viscosity coefficient and Khis the eddy diffusivityof heat. This closure scheme is often referred to as K theory.

The K theory has many limitations. Unlike the molecular viscosity coefficient,eddy viscosities depend on the flow rather than the physical properties of thefluid and must be determined empirically for each situation. The simplest modelshave assumed that the eddy exchange coefficient is constant throughout the flow.This approximation may be adequate for estimating the small-scale diffusion ofpassive tracers in the free atmosphere. However, it is a very poor approximationin the boundary layer where the scales and intensities of typical turbulent eddiesare strongly dependent on the distance to the surface as well as the static stability.Furthermore, in many cases the most energetic eddies have dimensions comparableto the boundary layer depth, and neither the momentum flux nor the heat flux isproportional to the local gradient of the mean. For example, in much of the mixedlayer, heat fluxes are positive even though the mean stratification may be very closeto neutral.

Page 22: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

:ws ¢¢( ) ( )zszss -=¢

( ) ( )( ) ( )

zsl

zslzszszss

¶¶

-=

--=-=¢

zsK

zswlws

¶¶

-=

¶¶¢-=¢¢

wlK ¢=

)-

)( -

Page 23: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

)(,xv

yuKvuuv

xuKuu hh ¶

¶+

¶¶

-=¢¢=¢¢¶¶

-=¢¢ 2

)(,xvK

zuKuwwu

yvKvv hh ¶

¶+

¶¶

-=¢¢=¢¢¶¶

-=¢¢ 2

)(,ywK

zvKvwwv

zwKww hh ¶

¶+

¶¶

-=¢¢=¢¢¶¶

-=¢¢ 2

���

ulKh ¢¢= ������������

Page 24: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

÷øö

çèæ

¶¶

+¶¶

¶¶

+

úû

ùêë

鶶

+¶¶

¶¶

+÷øö

çèæ

¶¶

¶¶

¶=

xwK

zuK

z

xv

yuK

yxuK

xxf

h

hhj

jxx

rrr

rr

rr

tr

1

)(1211

÷÷ø

öççè

涶

+¶¶

¶¶

+

÷÷ø

öççè

涶

¶¶

+÷÷ø

öççè

æ÷÷ø

öççè

涶

+¶¶

¶¶

¶=

ywK

zvK

z

yvK

yxv

yuK

xxf

h

hhj

jyy

r

rr

rr

tr

1

2111

÷øö

çèæ

¶¶

¶¶

+

÷÷ø

öççè

涶

+¶¶

¶¶

+÷øö

çèæ

¶¶

+¶¶

¶¶

¶=

zwK

z

ywK

zvK

yxwK

zuK

xxf hh

j

jzz

rr

rrr

rrr

tr

21

111

Page 25: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

2

2

2

2

2

2

zuK

yuK

xuKf hhx ¶

¶+

¶¶

+¶¶

=

2

2

2

2

2

2

zvK

yv

xvKf hy ¶

¶+÷÷ø

öççè

涶

+¶¶

=

2

2

2

2

2

2

zwK

yw

xwKf hz ¶

¶+÷÷ø

öççè

涶

+¶¶

=

KKK Vh ==

K

Page 26: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

÷÷ø

öççè

涶

¶¶

+÷÷ø

öççè

涶

¶¶

+÷÷ø

öççè

涶

¶¶

=z

kzy

kyx

kx hh

qrr

qrr

qrr

q 111*

hk��������������

�������������

k

� �

Page 27: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

���������

wvu ¢¢¢ ~~

zulu¶¶

-=¢

zulw¶¶

=¢zulwlK¶¶

=¢= 2

� ��

� ������ ��������������

Page 28: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23
Page 29: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

1 2 2 1 2 0 2

( ) 210 -Nmszx .~t 22 /1.0~ sms

zx÷÷ø

öççè

ært

smxp /10~1 3-

¶¶

r23 /10 sm

zzx -£÷÷ø

öççè

涶

rt

222 /10 smzx -£÷÷ø

öççè

æD

rt

zD %1010~ 1 =D -

zx

zx

tt

0»÷÷ø

öççè

涶

rt zx

z

Page 30: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

��== 2*uzx

rt

zul¶¶

=K 2

2*

2

u

zu

zul

zuKzx

=

¶¶

¶¶

=

¶¶

=rt

*uzul =¶¶

����*u

Page 31: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

*uzul =¶¶

zl k= k -

kzu

zu *=¶¶

0

* lnzz

kuu =

0z

Page 32: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

,**2 zuluzulK k==¶¶

=

��� �����

22* uCu Dzx rrt ==

2

0

2* ln/ ÷÷

ø

öççè

æ=÷

øö

çèæ=

zz

uucD k ������ �

Page 33: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23
Page 34: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

zypfu

zxpfv

zy

zx

¶¶

+¶¶

-=

¶¶

+¶¶

-=-

trr

trr11

11

������������������������������� ������ �����

January 27, 2004 9:4 Elsevier/AID aid

5.3 planetary boundary layer momentum equations 125

Fig. 5.3 Balance of forces in the well-mixed planetary boundary layer: P designates the pressuregradient force, Co the Coriolis force, and FT the turbulent drag.

means is needed to determine the vertical dependence of the turbulent momentumflux divergence in terms of mean variables in order to obtain closed equations forthe boundary layer variables. The traditional approach to this closure problem isto assume that turbulent eddies act in a manner analogous to molecular diffusionso that the flux of a given field is proportional to the local gradient of the mean. Inthis case the turbulent flux terms in (5.18) and (5.19) are written as

u′w′ = − Km

!∂u

∂z

"; v′w′ = − Km

!∂ v

∂z

"

and the potential temperature flux can be written as

θ ′w′ = − Kh

!∂θ

∂z

"

where Km(m2s− 1) is the eddy viscosity coefficient and Khis the eddy diffusivityof heat. This closure scheme is often referred to as K theory.

The K theory has many limitations. Unlike the molecular viscosity coefficient,eddy viscosities depend on the flow rather than the physical properties of thefluid and must be determined empirically for each situation. The simplest modelshave assumed that the eddy exchange coefficient is constant throughout the flow.This approximation may be adequate for estimating the small-scale diffusion ofpassive tracers in the free atmosphere. However, it is a very poor approximationin the boundary layer where the scales and intensities of typical turbulent eddiesare strongly dependent on the distance to the surface as well as the static stability.Furthermore, in many cases the most energetic eddies have dimensions comparableto the boundary layer depth, and neither the momentum flux nor the heat flux isproportional to the local gradient of the mean. For example, in much of the mixedlayer, heat fluxes are positive even though the mean stratification may be very closeto neutral.

Page 35: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

011=÷

øö

çèæ

¶¶

K¶¶

++¶¶

-zu

zfv

xp r

rr

011=÷

øö

çèæ

¶¶

K¶¶

+-¶¶

-zv

zfu

yp r

rr

2 31

( ) 02

2

=-+¶¶

gvvfzuK ( ) 02

2

=--¶¶

guufzuK

00 ==== wvuz ,

gg vvuuz ==¥® ,,

Page 36: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

( ) ( ) ( )gg ivuifivuifzivuK +-=+-

¶+¶2

2

( ) ( ))(

12

12

gg

zifzif

ivuBeAeivu +++=++

K-+

K

zf

ggg ezfvzfuuu K-

÷÷ø

öççè

æK

+K

-= 2

2sin

2cos

zf

ggg ezfvzfuvv K-

÷÷ø

öççè

æK

-+= 2

2cos

2sin

k

�������

)(,0 gg ivuBA +-==

Page 37: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

pnzf=

K2( )[ ]( )[ ]p

p

nng

nng

evv

euu-+

-+

-+=

-+=1

1

11

11

fz E

K==

2pd

- /-

410-

smc /10 25=K

scm /10 21-=n 1

1

Page 38: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

( )

g

g

g

g

gg

gg

zf

ggg

zf

ggg

z

z

uvuvvuvu

ezfvzfuv

ezfvzfuu

uvtg

-

+=

-+

=

÷÷ø

öççè

æK

-+

÷÷ø

öççè

æK

+K

-

=

=+

K-

K-

®

®

1

1

2cos

2sin

2sin

2cos

lim

lim

2

2

0

0

k

ba

( )bababatgtgtgtgtg

-+

=+1

����������� � ���4p

Page 39: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

January 27, 2004 9:4 Elsevier/AID aid

128 5 the planetary boundary layer

It can be shown that√

i = (1 + i) /√

2. Using this relationship and applying theboundary conditions of (5.28), we find that for the Northern Hemisphere (f > 0),A = 0 and B = −ug. Thus

u + iv = −ug exp!−γ (1 + i) z

"+ ug

where γ = (f/2Km)1/2.Applying the Euler formula exp(−iθ) = cos θ − isin θ and separating the real

from the imaginary part, we obtain for the Northern Hemisphere

u = ug

#1 − e−γ z cos γ z

$, v = uge−γ z sin γ z (5.31)

This solution is the famous Ekman spiral named for the Swedish oceanographerV. W. Ekman, who first derived an analogous solution for the surface wind driftcurrent in the ocean. The structure of the solution (5.31) is best illustrated by ahodograph as shown in Fig. 5.4, where the zonal and meridional components ofthe wind are plotted as functions of height. The heavy solid curve traced out onFig. 5.4 connects all the points corresponding to values of u and v in (5.31) forvalues of γ z increasing as one moves away from the origin along the spiral.Arrowsshow the velocities for various values of γ z marked at the arrow points. Whenz = π/γ , the wind is parallel to and nearly equal to the geostrophic value. It isconventional to designate this level as the top of the Ekman layer and to define thelayer depth as De ≡ π/γ .

Observations indicate that the wind approaches its geostrophic value at about1 km above the surface. Letting De = 1 km and f = 10−4 s−1, the definitionof γ implies that Km ≈ 5 m2 s−1. Referring back to (5.25) we see that for acharacteristic boundary layer velocity shear of |δV/δz| ∼ 5×10−3 s−1, this valueof Km implies a mixing length of about 30 m, which is small compared to the depthof the boundary layer, as it should be if the mixing length concept is to be useful.

Qualitatively the most striking feature of the Ekman layer solution is that, likethe mixed layer solution of Section 5.3.1, it has a boundary layer wind component

Fig. 5.4 Hodograph of wind components in the Ekman spiral solution. Arrows show velocity vectorsfor several levels in the Ekman layer, whereas the spiral curve traces out the velocity variationas a function of height. Points labeled on the spiral show the values of γ z, which is anondimensional measure of height.

Ekman������

Page 40: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23
Page 41: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

E

E

÷÷ø

öççè

涶

+¶¶

-=¶¶

yv

xu

zw dz

yv

xudz

zwE

E

ò ò ÷÷ø

öççè

涶

+¶¶

-=¶¶d

d

00

( ) ( )

( )

fK

efK

zdzKfeww

g

g

zKf

gE

2

12

2sin0

02

V

V

V

p

d

»

+=

=-¥

-

-

ò

Page 42: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

0!gV

S E2 3

f a2 ) - 1

D i

zwf

yv

xuf

dtd

¶¶

=÷÷ø

öççè

涶

+¶¶

-= 00z

( ) 0,;,0 ==== wDzwwz Ed

0!gV0!gV

Page 43: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

( ) gEg

Dfw

Df

dtd

Vdz

200

2K

-=-=

( ) ÷÷ø

öççè

æ K-= t

Df

gg 20

2exp0zz

K==

0

2f

Dt t

1

/ / 1 40f 410-

1 1 0 D0

dzzwf

yv

xuf

dtdD

g

E

ò úû

ùêë

鶶

=÷÷ø

öççè

涶

+¶¶

-=d

z00

Page 44: JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum equations 123 −f! u¯ − u¯g ∂v′w′ ∂z = 0 (5.19) where(2.23

6

42 .

1 3

3

5

E 3 5