312
DYNAMICAL OCEANOGRAPHY
◮
Example 13.1: Outcropping
Consider the situation for which
ˆ
w E (φ, θ)=
α
2
θ − θ 0
sin
2 θ
,
(13.61)
for a constant α =0on the domain N (θ), such that ˆ
w E (φ, θ 0 )=0and choose
h 2 (φ E ,θ 0 )=H 2 with H 2 constant. From (13.58) we then find that
γ 2 (h
2
2 (φ, θ) − H
2
2 )=α(θ 0 − θ)(φ E − φ).
(13.62)
Because φ E − φ>0,h 2 increases westward (see Fig. 13.4) when α>0. Because ∂h 2 /∂φ < 0, the meridional velocity v 2 < 0 (see (13.59b)) such that the
geostrophic transport is directed southward advecting water with a density ρ 2 .
In the case where α<0, and hence ˆ
w E > 0, then h 2 decreases westwards. If
for a certain latitude θ s
| α |
γ 2
(θ 0 − θ s )(φ E − φ W ) >H
2
2 ,
(13.63)
then h 2 becomes zero for a certain longitude φ s . This surfacing of a layer is
called ‘outcropping’. This simple example illustrates the problems that arise in
the ventilation theory when ˆ
w E > 0. In Fig. 13.4, the solution h 2 is plotted
Ex. 13.5
as a function of φ for different values of θ and for α>0. The slope ∂h 2 /∂φ
increases with southward and has a maximum at the southern boundary of the
domain θ = θ 2 which is the ‘outcropping’ curve of layer 1.
◭
As a next step, we consider the solution in the domain S(θ) where there are
three layers. The domain S(θ) is special in that layer 2 is no longer exposed to
the wind forcing. The conditions at the interfaces now become
p 1 − p 2 = γ 1 h 1
(13.64a)
p 2 − p 3 = γ 2 h 2 ,
(13.64b)
z = z 2 = −h 1 :
D
dt
(z + h 1 )=0,
(13.64c)
z = −z 3 = −(h 1 + h 2 ):
D
dt
(z + h 1 + h 2 )=0,
(13.64d)
with
γ 1 =
ρ 2 − ρ 1
ρ 0 ǫ p F p
; γ 2 =
ρ 3 − ρ 2
ρ 0 ǫ p F p
.
(13.65)
DYNAMICAL OCEANOGRAPHY
◮
Example 13.1: Outcropping
Consider the situation for which
ˆ
w E (φ, θ)=
α
2
θ − θ 0
sin
2 θ
,
(13.61)
for a constant α =0on the domain N (θ), such that ˆ
w E (φ, θ 0 )=0and choose
h 2 (φ E ,θ 0 )=H 2 with H 2 constant. From (13.58) we then find that
γ 2 (h
2
2 (φ, θ) − H
2
2 )=α(θ 0 − θ)(φ E − φ).
(13.62)
Because φ E − φ>0,h 2 increases westward (see Fig. 13.4) when α>0. Because ∂h 2 /∂φ < 0, the meridional velocity v 2 < 0 (see (13.59b)) such that the
geostrophic transport is directed southward advecting water with a density ρ 2 .
In the case where α<0, and hence ˆ
w E > 0, then h 2 decreases westwards. If
for a certain latitude θ s
| α |
γ 2
(θ 0 − θ s )(φ E − φ W ) >H
2
2 ,
(13.63)
then h 2 becomes zero for a certain longitude φ s . This surfacing of a layer is
called ‘outcropping’. This simple example illustrates the problems that arise in
the ventilation theory when ˆ
w E > 0. In Fig. 13.4, the solution h 2 is plotted
Ex. 13.5
as a function of φ for different values of θ and for α>0. The slope ∂h 2 /∂φ
increases with southward and has a maximum at the southern boundary of the
domain θ = θ 2 which is the ‘outcropping’ curve of layer 1.
◭
As a next step, we consider the solution in the domain S(θ) where there are
three layers. The domain S(θ) is special in that layer 2 is no longer exposed to
the wind forcing. The conditions at the interfaces now become
p 1 − p 2 = γ 1 h 1
(13.64a)
p 2 − p 3 = γ 2 h 2 ,
(13.64b)
z = z 2 = −h 1 :
D
dt
(z + h 1 )=0,
(13.64c)
z = −z 3 = −(h 1 + h 2 ):
D
dt
(z + h 1 + h 2 )=0,
(13.64d)
with
γ 1 =
ρ 2 − ρ 1
ρ 0 ǫ p F p
; γ 2 =
ρ 3 − ρ 2
ρ 0 ǫ p F p
.
(13.65)