Thermocline problem
307
bounded above by the ocean-atmosphere interface) has a constant density ρ 1 and
the lower layer (bounded below by a flat bottom) has a constant density ρ 2 >ρ 1 .
The layers are separated by a deformable interface at z = −h(φ, θ).
In each layer, the results of the homogeneous theory from the previous section
can be used; we denote the velocity vector in layer i with (u i ,v i ,w i ) T . In each
layer outside the Ekman layers, (13.7) applies and hence
v i sin θ =
1
cos θ
∂p i
∂φ
,
(13.43a)
u i sin θ = −
∂p i
∂θ
,
(13.43b)
∂p i
∂z
=0 ,
(13.43c)
1
cos θ
(
∂u i
∂φ
+
∂(v i cos θ)
∂θ
)+
∂w i
∂z
=0 .
(13.43d)
The boundary conditions at the thermocline (that is made dimensionless with
the depth D, i.e. h ∗ = Dh, but there is no aprioriscaling) z = −h(φ, θ) become
p 1∗ = p 2∗ ,
(13.44a)
D
dt
(z + h(φ, θ)) = 0 ⇒ u
∂h
∂φ
+ v cos θ
∂h
∂θ
+ w cos θ =0, (13.44b)
which represents the continuity of normal stress and the nonexistence of mass
transfer over the interface.
From (13.1a) and (13.44a) it follows that
γh = p 1 − p 2 ,
(13.45)
with γ =(ρ 2 − ρ 1 )/(ǫ p F p ρ 0 )=g ′ D/(2Ωr 0 U ), i.e. this parameter now contains
the reduced gravity g ′ = g(ρ 2 − ρ 1 )/ r ho 0 . From (13.45) it follows that small
pressure differences between layer 1 and 2 can cause substantial amplitudes in the
thermocline. Although the pressure is continuous over z = −h, this does not hold
Ex. 13.3
for the horizontal velocities (that are independent of z in each layer). It is not
difficult to show that the vertical velocity is continuous.
The final equations of the planetary two-layer model are obtained by integration of (13.43) over each layer. The vorticity equations then become (with
h 1 = h, h 2 =1− h),
v 1 h 1 cos θ =( ˆ
w E − w |z=−h )sinθ,
(13.46a)
v 2 h 2 cos θ =( w |z=−h − ˜
w E )sinθ,
(13.46b)
where ˆ
w E and ˜
w E are the vertical Ekman velocities at top and bottom of the
Ekman layer. Because of the continuity of w over the thermocline, the Sverdrup
307
bounded above by the ocean-atmosphere interface) has a constant density ρ 1 and
the lower layer (bounded below by a flat bottom) has a constant density ρ 2 >ρ 1 .
The layers are separated by a deformable interface at z = −h(φ, θ).
In each layer, the results of the homogeneous theory from the previous section
can be used; we denote the velocity vector in layer i with (u i ,v i ,w i ) T . In each
layer outside the Ekman layers, (13.7) applies and hence
v i sin θ =
1
cos θ
∂p i
∂φ
,
(13.43a)
u i sin θ = −
∂p i
∂θ
,
(13.43b)
∂p i
∂z
=0 ,
(13.43c)
1
cos θ
(
∂u i
∂φ
+
∂(v i cos θ)
∂θ
)+
∂w i
∂z
=0 .
(13.43d)
The boundary conditions at the thermocline (that is made dimensionless with
the depth D, i.e. h ∗ = Dh, but there is no aprioriscaling) z = −h(φ, θ) become
p 1∗ = p 2∗ ,
(13.44a)
D
dt
(z + h(φ, θ)) = 0 ⇒ u
∂h
∂φ
+ v cos θ
∂h
∂θ
+ w cos θ =0, (13.44b)
which represents the continuity of normal stress and the nonexistence of mass
transfer over the interface.
From (13.1a) and (13.44a) it follows that
γh = p 1 − p 2 ,
(13.45)
with γ =(ρ 2 − ρ 1 )/(ǫ p F p ρ 0 )=g ′ D/(2Ωr 0 U ), i.e. this parameter now contains
the reduced gravity g ′ = g(ρ 2 − ρ 1 )/ r ho 0 . From (13.45) it follows that small
pressure differences between layer 1 and 2 can cause substantial amplitudes in the
thermocline. Although the pressure is continuous over z = −h, this does not hold
Ex. 13.3
for the horizontal velocities (that are independent of z in each layer). It is not
difficult to show that the vertical velocity is continuous.
The final equations of the planetary two-layer model are obtained by integration of (13.43) over each layer. The vorticity equations then become (with
h 1 = h, h 2 =1− h),
v 1 h 1 cos θ =( ˆ
w E − w |z=−h )sinθ,
(13.46a)
v 2 h 2 cos θ =( w |z=−h − ˜
w E )sinθ,
(13.46b)
where ˆ
w E and ˜
w E are the vertical Ekman velocities at top and bottom of the
Ekman layer. Because of the continuity of w over the thermocline, the Sverdrup