JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum...
Transcript of JI NIE'S GROUP · January 27, 2004 9:4 Elsevier/AID aid 5.3 planetary boundary layer momentum...
����
�
�������
5
31
6.
2
2
4
3 %%
23 1 %- %
0 %
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.
�����
��
/
/ 5 . /
:
: ) (- 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
�������
第⼋章:湍流
聂绩
A = A+ A′
1
������� �������
sA
sA
BABAAB
BABA
A
AA
¶¶
=¶¶
¢¢+=
+=+
=¢
=
0
,
rr =
( )TRTRTRp
TTRpp
rrr
r
=¢+=
¢+=¢+ ,
���
� ��
! = #$%
����
( ) ( )[ ] ( )[ ] ( )[ ] 0=¶
¢+¶+
¶¢+¶
+¶
¢+¶+
¶¢+¶
zww
yvv
xuu
trrrrr
0)()()(=
¶¶
+¶
¶+
¶¶
+¶¶
zw
yv
xu
trrrr
0=¶
¢¶+
¶¢¶
+¶
¢¶zw
yv
xu )()()( rrr
������
�����
( ) ( ) úû
ùêë
é ¢¢-¶¶
+¢¢-¶¶
+¢¢-¶¶
++¶¶
-=- )(11 wuz
vuy
uux
Fxpvf
dtud
x rrrrr
( ) ( ) úû
ùêë
é ¢¢-¶¶
+¢¢-¶¶
+¢¢-¶¶
++¶¶
-=+ )(11 wvz
vvy
uvx
Fypuf
dtvd
y rrrrr
( ) ( ) úû
ùêë
é ¢¢-¶¶
+¢¢-¶¶
+¢¢-¶¶
++-¶¶
-= )(11 wwz
vwy
uwx
Fgzp
dtwd
x rrrrr
( ) ( ) ÷÷ø
öççè
æ ¢¢¶¶
+¢¢¶¶
+¢¢¶¶
+¶¶
+¶¶
+¶¶
=
÷÷ø
öççè
涢¶¢+
¶¢¶¢+
¶¢¶¢+
¶¶
+¶¶
+¶¶
=
¶¢¶¢+
¶¢¶¢+
¶¢¶¢+
¶¶
+¶¶
+¶¶
=
¶¶
+¶¶
+¶¶
)(1
1
wuz
vuy
uuxz
uwyuv
xuu
zuw
yuv
xuu
zuw
yuv
xuu
zuw
yuv
xuu
zuw
yuv
xuu
zuw
yuv
xuu
rrrr
rrrr
� ��������������������
j
jii xf
¶¶
=t
r1
÷÷÷
ø
ö
ççç
è
æ
¢¢-¢¢-¢¢-
¢¢-¢¢-¢¢-
¢¢¢¢-¢¢-=
÷÷÷
ø
ö
ççç
è
æ=
wwvwuwwvvvuvwuvuuu
zzyzxz
zyyyxy
zxyxxx
rrrrrrrrr
ttttttttt
t
���� �
2(� �0�
2(� !���'#���"�)!�3�!� ����!�0� �"�.�!
&�/���&�0!��*,���1%+�3$!�0�-0�
wuzx ¢¢-= rt
��������������� ���
�������
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)
Dθ
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
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)
Dθ
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
��������
��� �����
���!������ ���� ����������������
� ����������
( )i ji
j
u uf U
x¶
=¶
-
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.
:ws ¢¢( ) ( )zszss -=¢
( ) ( )( ) ( )
zsl
zslzszszss
¶¶
-=
--=-=¢
zsK
zswlws
¶¶
-=
¶¶¢-=¢¢
wlK ¢=
)-
)( -
)(,xv
yuKvuuv
xuKuu hh ¶
¶+
¶¶
-=¢¢=¢¢¶¶
-=¢¢ 2
)(,xvK
zuKuwwu
yvKvv hh ¶
¶+
¶¶
-=¢¢=¢¢¶¶
-=¢¢ 2
)(,ywK
zvKvwwv
zwKww hh ¶
¶+
¶¶
-=¢¢=¢¢¶¶
-=¢¢ 2
���
ulKh ¢¢= ������������
÷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
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
÷÷ø
öççè
涶
¶¶
+÷÷ø
öççè
涶
¶¶
+÷÷ø
öççè
涶
¶¶
=z
kzy
kyx
kx hh
qrr
qrr
qrr
q 111*
hk��������������
�������������
k
� �
���������
wvu ¢¢¢ ~~
zulu¶¶
-=¢
zulw¶¶
=¢zulwlK¶¶
=¢= 2
�
� ��
� ������ ��������������
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
��== 2*uzx
rt
zul¶¶
=K 2
2*
2
u
zu
zul
zuKzx
=
¶¶
¶¶
=
¶¶
=rt
*uzul =¶¶
����*u
*uzul =¶¶
zl k= k -
kzu
zu *=¶¶
0
* lnzz
kuu =
0z
,**2 zuluzulK k==¶¶
=
��� �����
22* uCu Dzx rrt ==
2
0
2* ln/ ÷÷
ø
öççè
æ=÷
øö
çèæ=
zz
uucD k ������ �
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.
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 ==¥® ,,
( ) ( ) ( )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 +-==
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
( )
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
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������
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
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
( ) 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
6
42 .
1 3
3
5
E 3 5