314
DYNAMICAL OCEANOGRAPHY
The potential vorticity q 2 is therefore conserved along streamlines.
At θ = θ 2 , we know the value of q 2 from the solution in the domain N (θ) and
we indicate it with q N
2 , i.e.,
q
N
2 =
sin θ 2
h 2
= −
sin θ 2
z 3
.
(13.71)
For θ<θ 2 , the potential vorticity q 2 is given by
q 2 =
sin θ
h 2
=
sin θ
z 2 − z 3
.
(13.72)
From (13.71-13.72) it follows that along a streamline in the second layer in the
domain S(θ),wehave
q 2 = q
N
2 ⇒ z 2 =(1−
sin θ
sin θ 2
)z 3 .
(13.73)
The Sverdrup balance (13.52) with ˜
w E =0, now becomes
(v 1 h 1 + v 2 h 2 )cosθ =sinθ ˆ
w E .
(13.74)
If the layer thicknesses are expressed in terms of z j , and the geostrophic velocities
and the relations (13.64a) are used with (13.74) we find
γ 1 z 2
∂z 2
∂φ
+ γ 2 z 3
∂z 3
∂φ
=sin
2 θ ˆ
w E (φ, θ),
(13.75)
which, after integration in the zonal direction, can be written as
γ 1 (z
2
2 (φ E ,θ) − z
2
2 (φ, θ)) + γ 2 (z
2
3 (φ E ,θ) − z
2
3 (φ, θ)) =
=2 s i n
2 θ
φ
φ E
ˆ
w E (φ, θ)dφ.
(13.76)
Because z 2 = −h 1 ,h 1 (φ, θ 2 )=0and u 1 (φ E ,θ)=0it follows from (13.52b)
that z 2 (φ E ,θ)=0 . The thickness of the upper layer remains zero at the eastern
boundary. If we write f =sinθ and f 2 =sinθ 2 and introduce h through
h 2 = h
f
f 2
,
(13.77)
then it follows from (13.73) that
h 2 = z 2 − z 3 = −
f
f 2
z 3 → h = −z 3 ; z 2 = −(1 −
f
f 2
)h,
(13.78)
such that (13.76) becomes
(γ 1 (1 −
f
f 2
)
2 + γ 2 )h
2 (φ, θ)=γ 2 z
2
3 (φ E ,θ) − 2sin
2 θ
φ E
φ
ˆ
w E (φ, θ)dφ. (13.79)