Thermocline problem
317
Additional Material
D: The material in the previous sections should (hopefully) make one prepared
for the more comprehensive texts on the ventilation theory of the thermocline
problem such as in sections 6.21 to 6.23 in Pedlosky (1987), chapter 4 in
Pedlosky (1996) and chapter 16 (sections 16.1 to 16.4) in Vallis (2006).
13.5.2. The internal boundary layer model
The second theory of the physics of the thermocline is the internal boundary
layer theory. The central idea of the internal boundary layer idea is that the thermocline arises through an advection/diffusion balance at mid-depth. The existence of such a balance can be studied by using the continuous equations (13.313.4) with the boundary conditions (13.5).
First we investigate whether we can extract a characteristic length scale for the
depth of the thermocline that depends both on the dynamics and the thermodynamics of the flow. Let W E be a (dimensional) characteristic vertical velocity due
to Ekman pumping at the surface, U = W E r 0 /D and ˆ
w E (φ, θ) the horizontal
(dimensionless) distribution of the Ekman vertical velocity. Now consider typical
horizontal density variations Δρ in the surface density field ρ s (φ, θ). The total
density ρ ∗ (φ, θ, 0) is then given by
ρ ∗ (φ, θ, 0) = ρ 0 +Δρρ s (φ, θ),
(13.88)
and with (13.1b) it follows that
ρ(φ, θ, 0) =
Δρ
ρ 0
gD
2ΩUr 0
ρ s (φ, θ)=
=
Δρ
ρ 0
gD 2
2ΩW E r 2
0
ρ s (φ, θ)=(
D
δ a
)
2 ρ s (φ, θ),
(13.89)
where
δ a =(
2Ω W E ρ 0
gΔρ
)
1/2 r 0
(13.90)
is called the advective thermocline scale. With W E =10 −6 ms −1 , Δρ/ρ 0 =10 −3
we find δ a ≈ 700 m which is a reasonable scale for the thermocline. The vertical
length scale δ a is the scale at which dynamically induced density differences are
balanced by changes in the surface density field. Another vertical length scale
is the diffusive scale δ D , that is defined by a balance between vertical advection
(w ∗ ∂ρ ∗ /∂z ∗ ∼ W E Δρ/δ D ) and vertical mixing (K V ∂ 2 ρ ∗ /∂z 2
∗ ∼ K V Δρ/δ 2
D ),
i.e.,
δ D =
K V
W E
.
(13.91)
317
Additional Material
D: The material in the previous sections should (hopefully) make one prepared
for the more comprehensive texts on the ventilation theory of the thermocline
problem such as in sections 6.21 to 6.23 in Pedlosky (1987), chapter 4 in
Pedlosky (1996) and chapter 16 (sections 16.1 to 16.4) in Vallis (2006).
13.5.2. The internal boundary layer model
The second theory of the physics of the thermocline is the internal boundary
layer theory. The central idea of the internal boundary layer idea is that the thermocline arises through an advection/diffusion balance at mid-depth. The existence of such a balance can be studied by using the continuous equations (13.313.4) with the boundary conditions (13.5).
First we investigate whether we can extract a characteristic length scale for the
depth of the thermocline that depends both on the dynamics and the thermodynamics of the flow. Let W E be a (dimensional) characteristic vertical velocity due
to Ekman pumping at the surface, U = W E r 0 /D and ˆ
w E (φ, θ) the horizontal
(dimensionless) distribution of the Ekman vertical velocity. Now consider typical
horizontal density variations Δρ in the surface density field ρ s (φ, θ). The total
density ρ ∗ (φ, θ, 0) is then given by
ρ ∗ (φ, θ, 0) = ρ 0 +Δρρ s (φ, θ),
(13.88)
and with (13.1b) it follows that
ρ(φ, θ, 0) =
Δρ
ρ 0
gD
2ΩUr 0
ρ s (φ, θ)=
=
Δρ
ρ 0
gD 2
2ΩW E r 2
0
ρ s (φ, θ)=(
D
δ a
)
2 ρ s (φ, θ),
(13.89)
where
δ a =(
2Ω W E ρ 0
gΔρ
)
1/2 r 0
(13.90)
is called the advective thermocline scale. With W E =10 −6 ms −1 , Δρ/ρ 0 =10 −3
we find δ a ≈ 700 m which is a reasonable scale for the thermocline. The vertical
length scale δ a is the scale at which dynamically induced density differences are
balanced by changes in the surface density field. Another vertical length scale
is the diffusive scale δ D , that is defined by a balance between vertical advection
(w ∗ ∂ρ ∗ /∂z ∗ ∼ W E Δρ/δ D ) and vertical mixing (K V ∂ 2 ρ ∗ /∂z 2
∗ ∼ K V Δρ/δ 2
D ),
i.e.,
δ D =
K V
W E
.
(13.91)