Wind-driven circulation
95
Figure 5.2. Local Cartesian coordinates as defined in (5.2).
After substitution of the scaled variables into (3.32) and use of the local coordinates (5.2), where derivatives are transformed as,
∂u ∗
∂φ
= U
∂u
∂x ∗
∂x ∗
∂φ
= U cos θ 0
r 0
L
∂u
∂x
,
(5.3)
we find the dimensionless equations
ǫ
Du
dt
+
L
r ∗
(δuw − uv tan θ)
− v
sin θ
sin θ 0
+ δw
cos θ
sin θ 0
=
−
cos θ 0
cos θ
r 0
r ∗
∂p
∂x
+
F
φ
I
Uf 0
, (5.4a)
ǫ
Dv
dt
+
L
r ∗
(δvw + u
2 tan θ)
+ u
sin θ
sin θ 0
=
−
r 0
r ∗
∂p
∂y
+
F θ
I
Uf 0
, (5.4b)
ǫδ
2 Dw
dt
− ǫδ
L
r ∗
(u
2 + v
2 ) − δu
cos θ
sin θ 0
=
−
∂p
∂z
+ δ
F r
I
Uf 0
,
(5.4c)
∂w
∂z
+2
D
r ∗
w −
L
r ∗
v tan θ ++
r 0
r ∗
∂v
∂y
+
r 0
r ∗
cos θ 0
cos θ
∂u
∂x
=0 ,
(5.4d)
with
D
dt
=
∂
∂t
+ u
r 0
r ∗
cos θ 0
cos θ
∂
∂x
+
r 0
r ∗
v
∂
∂y
+ w
∂
∂z
.
We use the identity
r ∗
r 0
=1+δ
L
r 0
z,
(5.5)