306
DYNAMICAL OCEANOGRAPHY
and the functions C i (θ) are determined from
ˆ
u
0 (0,θ)=−
1
sin θ
∂ ˆ
ψ 0
∂θ
(0,θ)=0
(13.39a)
lim
ζ→∞
ˆ
ψ
0 (ζ,θ)=C 1 (θ) = lim
φ→φ W
ψ
0 (φ, θ).
(13.39b)
At the eastern boundary the minus sign in (13.37) changes into a plus sign and,
as in the β-plane case, it follows that there is no boundary layer due to bottom friction. The function Ψ 0 (θ) in (13.34) has to be chosen such that the zonal velocity
is zero at φ = φ E and this fixes also C 1 and C 2 .
The boundary layer thickness δ(θ) is relatively small near the equator and increases monotonically with θ. The dimensional boundary layer thickness δ ∗ (θ)
follows from (13.37), the definition of ζ and the horizontal length scale L = r 0
and becomes
δ ∗ (θ)= ¯
E
1/2
V r 0 δ(θ)=
r 0
D cos 2 θ
(
A V sin θ
Ω
)
1/2 .
(13.40)
At θ =45 ◦ Nwefindthatδ ∗ ≈ 40 km for A V =10 −3 m 2 s −1 .
13.4. The planetary two-layer model
The governing equations in the stratified case are given by (13.4). When we neglect mixing and inertia in these equations, we find again the planetary geostrophic
and hydrostatic balances. Just as in the stratified quasi-geostrophic case in chapter 7, horizontal density differences will cause a vertical shear according to the
dimensionless thermal wind relations
sin θ
∂v
∂z
= −
1
cos θ
∂ρ
∂φ
,
(13.41a)
sin θ
∂u
∂z
=
∂ρ
∂θ
.
(13.41b)
Planetary thermal wind balance
On the planetary scale the thermal wind balance is
2Ω sin θ
∂v ∗
∂z ∗
= −
g
ρ 0 r 0 cos θ
∂ρ ∗
∂φ
,
2Ω sin θ
∂u ∗
∂z ∗
=
g
ρ 0 r 0
∂ρ ∗
∂θ
.
To help understand the existence of the thermocline one can idealize it as a
boundary between two layers of constant density. The upper layer (which is
DYNAMICAL OCEANOGRAPHY
and the functions C i (θ) are determined from
ˆ
u
0 (0,θ)=−
1
sin θ
∂ ˆ
ψ 0
∂θ
(0,θ)=0
(13.39a)
lim
ζ→∞
ˆ
ψ
0 (ζ,θ)=C 1 (θ) = lim
φ→φ W
ψ
0 (φ, θ).
(13.39b)
At the eastern boundary the minus sign in (13.37) changes into a plus sign and,
as in the β-plane case, it follows that there is no boundary layer due to bottom friction. The function Ψ 0 (θ) in (13.34) has to be chosen such that the zonal velocity
is zero at φ = φ E and this fixes also C 1 and C 2 .
The boundary layer thickness δ(θ) is relatively small near the equator and increases monotonically with θ. The dimensional boundary layer thickness δ ∗ (θ)
follows from (13.37), the definition of ζ and the horizontal length scale L = r 0
and becomes
δ ∗ (θ)= ¯
E
1/2
V r 0 δ(θ)=
r 0
D cos 2 θ
(
A V sin θ
Ω
)
1/2 .
(13.40)
At θ =45 ◦ Nwefindthatδ ∗ ≈ 40 km for A V =10 −3 m 2 s −1 .
13.4. The planetary two-layer model
The governing equations in the stratified case are given by (13.4). When we neglect mixing and inertia in these equations, we find again the planetary geostrophic
and hydrostatic balances. Just as in the stratified quasi-geostrophic case in chapter 7, horizontal density differences will cause a vertical shear according to the
dimensionless thermal wind relations
sin θ
∂v
∂z
= −
1
cos θ
∂ρ
∂φ
,
(13.41a)
sin θ
∂u
∂z
=
∂ρ
∂θ
.
(13.41b)
Planetary thermal wind balance
On the planetary scale the thermal wind balance is
2Ω sin θ
∂v ∗
∂z ∗
= −
g
ρ 0 r 0 cos θ
∂ρ ∗
∂φ
,
2Ω sin θ
∂u ∗
∂z ∗
=
g
ρ 0 r 0
∂ρ ∗
∂θ
.
To help understand the existence of the thermocline one can idealize it as a
boundary between two layers of constant density. The upper layer (which is