298
DYNAMICAL OCEANOGRAPHY
13.2. Formulation of the problem
We consider an ocean basin that is bounded in the zonal direction by continents
at φ = φ W (θ) and φ = φ E (θ) and which has a meridional extent [θ S ,θ N ].I nt h e
vertical direction, the ocean water is bounded by an ocean bottom with average
depth D and by the ocean-atmosphere interface with a mean position at z =0.
On the planetary scale, the characteristic length scale of the flow is L = r 0 and
because the density variations are large, the static stability parameter (the buoyancy frequency N ) will vary strongly. Hence we can no longer use a reference
density ¯
ρ ∗ (z) and the only reasonable choice for such a reference is a constant
¯
ρ ∗ = ρ 0 . With these choices, the appropriate scales for pressure and density
follow as (cf. section 3.3)
p ∗ = −gDρ 0 z +2ρ 0 U Ωr 0 p,
(13.1a)
ρ ∗ = ρ 0 (1 +
2ΩUr 0
gD
ρ)=ρ 0 (1 + ǫ p F p ρ),
(13.1b)
where z ∗ = Dz.H e r eǫ p = U/(2Ωr 0 ) and F p =4 Ω 2 r 2
0 /(gD) are the planetary
Rossby and rotational Froude numbers.
To simplify matters, we will take a linear equation of state
ρ ∗ = ρ 0 (1 − α T (T ∗ − T 0 )+α S (S ∗ − S 0 )),
(13.2)
with constant α T and α S . Because we will focus on the understanding of the vertical structure of the wind- and density driven flow, we neglect horizontal mixing
of momentum, heat and salt. The vertical mixing of heat and salt is represented
by the vertical diffusivity K V .
With a linear equation of state the equations for temperature and salinity (3.32ef) can be combined into one equation for the density ρ, which becomes (in spherical coordinates)
Dρ
dt
=
u
cos θ
∂ρ
∂φ
+ v
∂ρ
∂θ
+ w
∂ρ
∂z
= λ V
∂ 2 ρ
∂z 2 .
(13.3)
with λ V = K V r 0 /(UD 2 ). The parameter λ V represents the small scale vertical
mixing of density. With an estimate of K V =5× 10 −5 m 2 s −1 and U =1 0 −3
ms −1 it follows that λ V ≈O(1), but this value is very uncertain. Likely, the value
of λ V is much smaller.
From chapter 3, the scaled momentum balance and the continuity equation can
(in the limit δ = D/r 0 → 0) be written as
ǫ p
Du
dt
− uv tan θ
− v sin θ = −
1
cos θ
∂p
∂φ
+
1
2
¯
E V
∂ 2 u
∂z 2 , (13.4a)
ǫ p
Dv
dt
+ u
2 tan θ
+ u sin θ = −
∂p
∂θ
+
1
2
¯
E V
∂ 2 v
∂z 2 ,
(13.4b)
DYNAMICAL OCEANOGRAPHY
13.2. Formulation of the problem
We consider an ocean basin that is bounded in the zonal direction by continents
at φ = φ W (θ) and φ = φ E (θ) and which has a meridional extent [θ S ,θ N ].I nt h e
vertical direction, the ocean water is bounded by an ocean bottom with average
depth D and by the ocean-atmosphere interface with a mean position at z =0.
On the planetary scale, the characteristic length scale of the flow is L = r 0 and
because the density variations are large, the static stability parameter (the buoyancy frequency N ) will vary strongly. Hence we can no longer use a reference
density ¯
ρ ∗ (z) and the only reasonable choice for such a reference is a constant
¯
ρ ∗ = ρ 0 . With these choices, the appropriate scales for pressure and density
follow as (cf. section 3.3)
p ∗ = −gDρ 0 z +2ρ 0 U Ωr 0 p,
(13.1a)
ρ ∗ = ρ 0 (1 +
2ΩUr 0
gD
ρ)=ρ 0 (1 + ǫ p F p ρ),
(13.1b)
where z ∗ = Dz.H e r eǫ p = U/(2Ωr 0 ) and F p =4 Ω 2 r 2
0 /(gD) are the planetary
Rossby and rotational Froude numbers.
To simplify matters, we will take a linear equation of state
ρ ∗ = ρ 0 (1 − α T (T ∗ − T 0 )+α S (S ∗ − S 0 )),
(13.2)
with constant α T and α S . Because we will focus on the understanding of the vertical structure of the wind- and density driven flow, we neglect horizontal mixing
of momentum, heat and salt. The vertical mixing of heat and salt is represented
by the vertical diffusivity K V .
With a linear equation of state the equations for temperature and salinity (3.32ef) can be combined into one equation for the density ρ, which becomes (in spherical coordinates)
Dρ
dt
=
u
cos θ
∂ρ
∂φ
+ v
∂ρ
∂θ
+ w
∂ρ
∂z
= λ V
∂ 2 ρ
∂z 2 .
(13.3)
with λ V = K V r 0 /(UD 2 ). The parameter λ V represents the small scale vertical
mixing of density. With an estimate of K V =5× 10 −5 m 2 s −1 and U =1 0 −3
ms −1 it follows that λ V ≈O(1), but this value is very uncertain. Likely, the value
of λ V is much smaller.
From chapter 3, the scaled momentum balance and the continuity equation can
(in the limit δ = D/r 0 → 0) be written as
ǫ p
Du
dt
− uv tan θ
− v sin θ = −
1
cos θ
∂p
∂φ
+
1
2
¯
E V
∂ 2 u
∂z 2 , (13.4a)
ǫ p
Dv
dt
+ u
2 tan θ
+ u sin θ = −
∂p
∂θ
+
1
2
¯
E V
∂ 2 v
∂z 2 ,
(13.4b)