Wind-driven circulation
97
5.2.2. Boundary conditions
The only boundary conditions which need further analysis are the conditions
(3.37) at the ocean-atmosphere boundary. The mean interface position is at z ∗ =0
and from (3.37d) and (5.1) it follows that at z ∗ = h ∗ (or z = h ∗ /D):
−gh ∗ ρ 0 + ρ 0 Uf 0 Lp=0,
(5.9)
where p a∗ =0is chosen as reference level for the pressure.
To determine the scaling of the interface amplitude, we write h ∗ /D =
μη(φ, θ, t), where μ is, for the moment, still unknown. For small deviations
h ∗ /D, it follows that at z =0:
0=−gDρ 0 μη + ρ 0 Uf 0 Lp |z=0 + ··· → μη =
Uf 0 L
gD
p,
(5.10)
This shows that μ = Uf 0 L/(gD)=ǫF , where F = f 2
0 L 2 /(gD) is the Froude
number (cf. section 3.1.5). For midlatitude flows F = O(1), see Table 5.3 and
hence ǫF ≪ 1 justifying the expansion (5.10). As a consequence, we write
z = ǫF η(φ, θ, t) and the normal stress balance (3.37d) obtains the simple dimensionless form
p = η
(5.11)
With τ 0 as a characteristic value of the wind stress and hence τ ∗ = τ 0 τ ,t h e
tangential stress boundary conditions become
τ 0 D
ρ 0 A V U
τ
φ =
∂u
∂z
+
A H
A V
D
r ∗
δ
cos θ
∂w
∂φ
,
(5.12a)
τ 0 D
ρ 0 A V U
τ
θ =
∂v
∂z
− δ
A H
A V
D
r ∗
∂w
∂θ
.
(5.12b)
Although the values of the mixing coefficients A H and A V are not well known,
the second terms in the right hand side are (with plausible estimates) much smaller
than the first ones and they can be neglected in the limit δ → 0.
Finally, the kinematic condition in (3.37a) is written as
w = ǫF
∂
∂t
+
L
r ∗
u
cos θ
∂
∂φ
+ v
∂
∂θ
η,
(5.13)
and in local coordinates (5.2) with L/r 0 ≪ 1 and δ → 0, this equation becomes
w = ǫF
∂
∂t
+ u
∂
∂x
+ v
∂
∂y
η,
(5.14)
with η = η(x, y, t).