u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
1
u? ? `D vè0??
?Bc v H?a
? v H?a# +?
? H?a??
?ú￥M??
p
? =)ì
@ a & L N B O ?èh D
? & L N B O
?? 3 J D I B S E T P O
?
×? H?a??
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p & L N B O ?èh D b
?1 v H?a# +?
1 v ￥? ?sa
1.1 v H?a￥?l D1V°¤¤(¨?1 1-1.5km a μu
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Boundary Layer) b
1.2 PBL 8sa
?m 1.1,s1 ?a :
m 1.1 v H?asa
Uim
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
2
m 1.2 v ? ?sa
òaèn￥ a?]￥ ?
v H?a ??H?aH?a ?a
L1a V
?a
í1a ¤1a1
?H?aèY
a SL(Surface Layer)
? X
￡
Ekmana
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? ?a
2 L1a￥?1+?
s0T? ? ?1T¨ ?1??
T s0 ? ? b
3 í1a￥?1+?
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f ? V
8
{ b
2).?_+??
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?
?-9F b
?
?
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V
z
D V
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3). t ?
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,
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>>
? ??
b
5).u
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üA1 MoT¨ <
??? y??è b
4 Ekman a￥?1+?
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1).u
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f ?]?×1 b
2).t ?
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? b
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p b
5 1?v a￥?1+?
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{
￡ ü a0 ?? S
f ? ?1T¨ b
2).
s??H?a?<°?￥?Y /H?Hq'1v H?a￥
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()
e
e
h
zh
ω ω
=
=
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p
1 í1a??
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p
èY
a?t ?
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??
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?
? v? ?
0
z
z
TT==
nullnull null
è
O
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m 1.3 L. Prandtl (1875-1953)￡SH?a#u
@DE
? z
0
?11
?Y? ?l1?
?1
,￥ú ?ó
L ?? ′1-?/?
?Y?t ü (ú￥ 1/30 b
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
4
m 1.4 ?]/?
?￥Y?
? ? μ+
Prandtl ?é ?
?
zT
z
V
k
z
ρ
?
=
?
nullnull
nullnull
1.2
z
k 1<°u
@"
? b SL??_?
? zM?5
0
zz
V
kT
z
ρ
?
=
?
=è
?
1.3
?
z
k
? z7M?L?
2
z
V
kl
z
?
=
?
nullnull
=
2
V
l
z
?
?
1.4
l1 ? μ+?é b5
0
2
2
Z
V
lT
z
ρ
?
??
=
??
?
??
1.5
0
*
Z
T
V
lV
z ρ
?
==
?
1.6
í1a? ?
?
*
V =
è
? l ≠è
?Dú z?v a2×??μ1 b
1.1 ??a2/￥
?s??
p
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
5
??a2?u
@? %?D/?
?￥? ?T¨ b ?/?
? í lü l b PrandtlL? l
^ z￥L?
f
? lkz= k
^ 5
￡
Von Karmanè
?
0.35 j0.42B? | 0.4
m 1.5 T. von Karman (1881-1963)
?S÷ ￥ ?D#
@8? ?DE
∴
*
V
kz V
z
?
=
?
1.7
*
1VVdV
zdzkz
?
==?
?
1.8
?¨/H?Hq
0
0
zz
V
=
=
∴
*
0
Vz
VLn
kz
=
0 s
zzh≤ ≤
1.9
i i1 ?
?s??
p
T< 1 cha.1 i4
y¤ 2Df
??m èq Graphmatica
U :°?US"?
?US"??￥
?s?
p
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
6
0.25π 0.5π 0.75π π 1.25π 1.5π 1.75π 2π 2.25π 2.5π 2.75π 3π
2
4
6
8
10
12
14
V
z
1 2 3 4 5 6 7 8 9
0.1
1
10
V
ln
z
m 1.6 °?US"?
?US"??￥
?s?
p
1.2 d??a2/￥·
?s??
p
LaixtmanL?
1
lz
ε
β
?
=
1.10
&1???
? ε 1a2?
??×?a2 10ε? <<??a2 0ε = ×?a2 01ε< < b
*
V
lV
z
?
=
?
∵
1.11
*
1
VV
zz
ε
β
?
?
∴ =
?
1.12
!
0
zz=
H ˉ
?@
?s??
p
0
*
0
zz
VV
zkz
=
?
=
?
? ∵
0
*
1
0
zz
VV
ε
β
=
?
?
=
?
∴
0
kz
ε
β =
1.13
0
z
lkz
z
ε?
??
=
??
??
1.14
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
7
∴
1
1
0**
00 0
ε
ε
?
?
??
? ??
==
?? ??
?
??
??
zVVVz
zkzz kzz
1.15
?¨
0
0
ZZ
V
=
= ,
*
0
1
Vz
V
kz
ε
ε
??
??
??∴ =?
??
??
??
0 s
zzh≤≤
1.16
i i
a·
?s??
p
y¤ 2Df
??m èq Graphmatica
U :°?US"?í1ad??a2/?
?ú￥s?
z
?3π ?2.5π ?2π ?1.5π ?π ?0.5π 0 0.5π π 1.5π 2π 2.5π 3π
2
4
6
8
10
12
14
16
18
20
×?
?×?
m 1.7 í1a ü (?
?ú￥M?
1 ×?a2 2 ??a2 3 ?×?a2
?
1.16
T ?a2 t???' 0ε → μ
0
** *
00 0
00 0
1
lim lim lim ln ln
z
z
VV Vzz z
V
kkzzk
ε
ε
εε ε
ε
→→ →
??
??
???
??
??
??
===
??
??
??
??
??
1.17
a·
?s??
p (1.16) ??? →
??
1 ?
?s??
p (1.9)
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
8
í1a? ￡
ê? a
￡
{
?
1
A
? z￥M??
pD?
{
?
?
? ?
? b
XD :í1a??_?
?ú?M?
?
?ú￥M??C
?
??a2·
??
p
d??a2
ù5 Ekmana??_
^?
?ú?M$?
?
?ú￥M????
p$
m 1.8 V. W. Ekman (1874-1954) ??Z DEZ í
@?0
@ ?
?￥í |
2 Ekmana??
?ú￥M??
p
2.1 Ekman
?L3 (Ekman spiral)
? - V? Ekman a?v ?
?@ o Ekman ü? p ?¨ -
?￥ Ekman a￥?1+? 1 a2
L? ??è a ü
@8? ?
dL?[M? S
f ? V-
{ a
￡ ü a0 ??
?ú?M5μ
Ekman a
v ?Z?F
11
z
p u
fv k
x zz
ρ
ρρ
???
??
?=? +
??
?? ?
??
1.18
11
z
p v
fu k
yz z
ρ
ρρ
???
??
=? +
??
?? ?
??
1.19
?
! ρ =è
?
z
k =è
? f=è
?i? ?1?
T
11
,
gg
p p
uv
f yfxρρ
? ?
=? =
? ?
5
1.18 a
(1.19)
T V?1
()
2
2
0
zg
u
kfvv
z
?
+?=
?
1.20
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
9
{
()
2
2
0
zg
v
kfuu
z
?
??=
?
1.21
|H?Hq /H? z=0 0u → , 0v→
H? z →∞
g
uu→ ,
g
vv→
?D
??
ì a7t ?
? ? V ?31 ? 71V
?@ú￥ú
L=?M?
1?
?¨ ˉ
?E
7
*
Vuiv=+a ? Z??¨
6¨ aBQ?3 u ? v b `
￥?¨ Ekman
?
L3 aé?￥8???3 a l p3
1.20 + i
1.21 V¤
()()()
2
2
zgg
k u iv if u iv if u iv
z
?
+? +=? +
?
1.22
1 p3ZL | xà ü??aL5 0, 0
g
p
v
x
?
==
?
'N
H1?oμê_s
μ
2*
*
2zg
V
k ifV ifu
z
?
?=?
?
1.23
2*
*
2zg
dV
k ifV ifu
dz
?=?
1.23
’
Z?￥?é Bí=Qd
Qè±sZ?
p3ZE d
QY3"
QY3
μ ???
T$d
Q+3
43E a
k3E
T< h]D
ì [a1-
p3Z?
1.23
’
￥ 8V? b
3Z?
1.23
’
V¤
11
22
*
exp exp
g
zz
if if
VuivA zB zu
kk
?? ? ?
?? ??
?? ? ?
=+= + ? +
?? ??
?? ??
?? ? ?
?? ? ?
1.24
y1
1
2
i
i
+
= i?H?Hq V?"
? A =0
g
B u=?
( )1* iz
g g
Vuivue u
γ?+
∴ =+=? +
1.25
?
1
2
2
z
f
k
γ
??
=
??
??
b ?¨ˉ
??￥ x ?
Euler
T cos sin
i
ei
θ
θ θ
?
=? V¤
()
1cos
z
g
uu e z
γ
γ
?
=?
sin
z
g
vue z
γ
γ
?
= ()
s e
hzh≤ ≤
1.26
i iEkmana?
?ú￥s?3
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
10
y¤ 2Df
??m èq Graphmatica
U :Ekmana?￥ u
?L ,v
x￡Ls
?ú￥s?
z
?3π ?2.5π ?2π ?1.5π ?π ?0.5π 0 0.5π π 1.5π 2π 2.5π 3π
2
4
6
8
10
12
14
16
18
20
v
u
Ekmana?￥ u,vs
?ú￥s?
y¤ 2D?
??f
??m èq Mathgv
U Ekman
?L
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
11
m 1.9 Ekmana?
?ú￥M?
m 1.10 ?? Ekman
?L
′L ü?′ ??L ??′
LL 4?′
2.2 Ekmana# Ekman
?L￥?1+?
Ekman
? L : Ekmana??
?
O
￥L Ekmana??]ú￥?
?
O
￥ ?L b
1 Ekman ú
e
zh
π
γ
==
H
( )
11.04,0
gg
ueu uv
π?
=+ ≈ =∴Nú
,??aL8?
?
{v?1?
?
1 ;? z →∞
H ,0
g
uuv==(M11? ) b
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
12
y¤ Matlab SD/
??y9
? :e
π?
≈0.0432
?
e
h 1 Ekman ú
0?ú ?_D?aL?BQ×?￥ú;?_D1?Z_M]￥
ú7?
21
z
k
fγ
= 1 Ekman Sú b
2 Ekman ü?
Ekman a ? ? ü?/?μ·_?aB§s
G→D bi ?úa·_?a￥t ?
{
?
?
?
N
H a0 ?T??·h ?{
? b???
^ ?
ü?7 O9
^ ?￥ ü?￥1 p
?m 1.6
?
U b
m 1.11 Ekmana? ? ? ü?/￥ b ?
3) 1
??D?aL￥C?
1
?
/H?) ?D?aL￥C?r?
Kv′ 45
null
b
000
sin
lim lim lim 1
1cos
z
z
zzz
vez
tg
uez
γ
γ
γ
α
γ
?
?
→→→
== =
?
∵
1.27
4
π
α∴ =
(0,)
2
π
α
? ?
∈
? ?
? ?
4
?LVe?
0z =
H 0uv==
?[
?LüV u-v ü
?￥e? b
5 Ekman
?L1B??
?L
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
13
?L￥ MLZ_
￡ ü?￥<° MM
V
z
?
?
nullnull
D1??
L=?￥
O
μ￥C? ò?úB"?
?? 45
°
b
D S?- aMo\é è
Ekmana ?
? Z →v
Rossbyéoo? ?
? v →Z
Lorenz§ (Chaos) ?
? v
???d? ù
@ →1 ? aü6 a
? a l
T< 2 cha.1 i11
?3 Ekman =)ì
@?èh D
1. Ekman
-
! ρ =è
?
(é? Vav '
0
d
dt
ρ
=
1.28
5 ??Z?1
0
uvw
xyz
? ??
+ +=
???
1.29
∴
()()
1cos 1cos
gzz
g
u
u
ue z e z
x xx
γγ??
?
??
??
=? =?
??
?? ?
=
()
1
1cos
z
p
ez
x fy
γ
γ
ρ
?
????
??
??
??
??
= i
()
1
1cos
z
p
ez
f yx
γ
γ
ρ
?
????
?
??
??
??
=0
∴
wv
zy
??
=?
??
1.30
00
ee
hh
wv
dz dz
zy
??
=?
∫∫
1.31
/H?) , z=0,w=0
?[
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
14
w
0
()
e
h
e
hvdz
y
?
? ?
=?
? ?
?
∫
=
()
0
sin
e
h z
g
ue zdz
y
γ
γ
?
?
??
?
??
?
∫
=
0
sin
ee
z
hh
g
e
z
ue dz
yh
π
π
?????
?
? ????
?
??
??
∫
=
()
1
1
2
ge
uh e
y
π
π
?
? ??
??+
??
?
??
2
g
e
u
h
yπ
?
≈?
?
(e
π?
≈0.0432)
1.32
ggg
g
u
vu
x yy
?
?
??
=?=?
?? ?
∵
1
2
()
22
e z
eg g
h k
wh
f
? ?
π
??
∴ ==
??
??
1.33
i iEkman
T
t ?il
1 VT1 Ekmana
H?Hq1?v /H?Hq b
2
^ PBL D1?v ￥ ó" b PBL ?￥<°
?D1?v ?￥???11?v i ?è
0
g
? ≠ i OH?aμu
@ ?T¨
H
0
z
k ≠ PBL?Aμ () 0
e
wh ≠ b
3B??1 Ekman
T
Ekman
?
T b
2=)ì
@?èh D
2.1=)ì
@ (Secondary circulation)
1) ?l ?1?v ??èì
@?H?au
@ ?T¨7á
3￥ ,YF ?èì
@
i
s ??
￥<°ì
@ ? b
2) ?? 1?v ? è aQ èì
@?￥1?v ?H?aW￥
6?/??
=)ì
@￥<°s| F-??é
?êT¨á
3￥
￡ ü??? ?
@
=)ì
@￥
￡ ü
s|? b
3) il ￡
U
H?aD1?v -Wt ?
?D￥?1o? b u
@
{
?
? ?
¤V?t
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15
?
<°?D￥Q1o? =)ì
@ yV?r
qút ?
<°?D￥?1o? b
m 1.12 ?av ?￥=)ì
@
2.2 Ekman (Ekman pumping)
YV=)ì
@1?v DH?a Vé?é
?
?t ?
? ￡
a
￡s￥?D1?
v ??
v￥ b YV () 0
e
wh < $ ?H?a
6BZ
?H?a??
l￥5$ ?1?
v bü???H?au
@ ?T¨á
3￥<°??1 Ekman
=)ì
@￥<°s|
]
H? ()
e
wh 1 Ekman
? b
2.3 èh D (Spin-down)
1 ?l YV Ekman H?aD1?v á
3é
??
?D
P1?v ?h DM?
￥?è <
g
?
1?hl?1èh D
6BZ
?H?a
??á
3èF <???
u
@ ?T¨
P? ù￥èF <$·h b
2èh D
T
!v (é? Va
ρ =è
? ??Z?1
0
uvw
xyz
? ??
+ +=
???
(1.34)
??
v?è
?@￥e?a￥?Z?
()
()
df
uv uv w
fff
dt x y x y z
ζ
ζ
+ ????
? ????
=? + + ≈? + =
????
?
????
(1.35)
?
!
g
ζ ζ≈
u? ? `D vè0?? I÷ a?? ???y?Dyv SD" ùS ü?
q ?T
@ a ùS ü
16
∴
()
g
df
w
f
dt z
ζ +
?
=
?
(1.36)
1?v ?￥
@aé?<°
ú s
VH?a??
@a? i
! z" H ()0wH=
5μ
()
()
0
eee
HH
g
hhwh
df
w
dz f dz f dw
dt z
ζ +
?
==
?
∫∫∫
(1.37)
?¨ fè
?
ü
?í
?' f=è
? 0
df
dt
= 5
()
H
g
e
he
d
dz fw h
dt
ζ
=?
∫
(1.38)
y1 0
g
z
ζ?
=
?
g g
g
vu
x y
ζ
??
=?
??
∵ ρ =è
?
0
gg
vu
zz
? ?
→==
??
'1???
?ú z
?M ,5
(1.39)
HWs
V 0→ts
!
00g tg
ζζ
=
=
() () ()
1
2
2
2
1
g eee
z
g
ee
d fw h fw h fw h
fk
Ekman pumping
hdt H h H H
H
H
ζ
ζ
??
=? =? ≈? ?
??
? ??
??
?
??
??
(1.40)
1
2
2
2
0
z
fk
t
H
gg
eζζ
??
????
?
??
??
??
∴ = (1.41)
i ièh D
T
?lèh D
HW
e
τ ú1 H￥?a?è??èh D
P <hl?
S′￥ 1/e
H
?3
1￥
HW'
1
2
2
e
z
H
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