310
DYNAMICAL OCEANOGRAPHY
θ
z
ρ
1
ρ
2
z 3
z
2
h 2
h 2
h 1
v 2
v 2
v 1
z
4
= - 1
φ
z
1
= 0
θ
0
θ
2
θ
1
N( )
θ
S( )
θ
h 3
Figure 13.3. Sketch of the three-layer ocean model, where the thickness of the layer with density
ρ1 becomes zero at θ = θ2; note that θ2 <θ0.
The vertical velocity ˆ
w E due to Ekman pumping is assumed negative on the whole
interval θ 1 <θ<θ 0 and ˆ
w E (θ = θ 0 )=0 . In the domain S(θ), for which
θ 1 <θ<θ 2 , layer 1 is exposed to the wind forcing. In the domain N (θ) for
which θ 2 <θ<θ 0 , layer 2 surfaces.
We first consider the domain N (θ) where h 1 =0. In each layer j the governing
equations of the flow outside the Ekman layer are
v j sin θ =
1
cos θ
∂p j
∂φ
,
(13.52a)
u j sin θ = −
∂p j
∂θ
,
(13.52b)
∂p j
∂z
=0 ,
(13.52c)
∂u j
∂φ
+
∂(v j cos θ)
∂θ
+cosθ
∂w j
∂z
=0 .
(13.52d)
Just as in the two-layer model in the previous section, the pressure and vertical
velocity are continuous over the interfaces and hence
p 2 − p 3 = γ 2 h 2 = −γ 2 z 3 ,
(13.53a)
z = z 3 = −h 2 :
D
dt
(z + h 2 )=0,
(13.53b)