Adjustment
205
closed by specifying kinematic boundary conditions at the basin boundaries and
the initial conditions
t =0:Ψ=0.
(9.35)
In dimensional quantities (note that time, streamfunction and horizontal coordinates were scaled with 1/(β 0 L), UL and L) the equation (9.35) becomes
∂
∂t ∗
(∇
2
∗ ψ ∗ − λ 0 ψ ∗ )+β 0
∂ψ ∗
∂x ∗
=
1
ρ 0 H 1
∇.(T ∗ ∧ e 3 ),
(9.36)
where
λ 0 =
f 2
0
g′
(
1
H 1
+
1
H 2
)=
1
L 2
D1
,
in the baroclinic response and λ 0 =0in the barotropic case. Note that from
section 5.3, the adjustment problem for the constant density case satisfies also
an equation of the form (9.34) but then with Λ=F (and λ 0 = f 2
0 /(gD)). Often a linear damping (−ε 0 ∇ 2
∗ ψ ∗ ) is added to the right hand side with a damping
coefficient ε 0 (s −1 ).
The equation (9.34) provides insight into the response of the flow to different
frequencies in the wind forcing. Consider the baroclinic case with H 2 ≫ H 1 and
hence F 2 ≪ F 1 . We can write
Λ=F 1 =(
L
L D1
)
2
(9.37a)
L
βUτ p
=
1
β 0 Lτ p
= τ β /τ p
(9.37b)
L
βUτ p
Λ=
L
β 0 L 2
D1 τ p
=
L
c r τ p
,
(9.37c)
where c r = β 0 L 2
D is the magnitude of the Rossby phase speed. The factor L/c r
is hence a wave time scale which is called the adjustment time scale τ c .
Ex. 9.3
With τ p being the time scale of variation of the wind forcing, the following
cases can be distinguished
(i) High frequency forcing, i.e., τ β /τ p ≫ 1 and τ c /τ p ≫ 1. In this case, fluctuations in the wind will not generate Rossby waves and the response is local.
(ii) Low frequency forcing, i.e., τ β /τ p ≪ 1 and τ c /τ p ≪ 1. In this case, the first
term in (9.34) can be neglected and the ocean is always in Sverdrup balance
with the changing wind stress.
(iii) On time scales τ p on the order of τ c or/and τ β Rossby waves are an important
component of the response.
205
closed by specifying kinematic boundary conditions at the basin boundaries and
the initial conditions
t =0:Ψ=0.
(9.35)
In dimensional quantities (note that time, streamfunction and horizontal coordinates were scaled with 1/(β 0 L), UL and L) the equation (9.35) becomes
∂
∂t ∗
(∇
2
∗ ψ ∗ − λ 0 ψ ∗ )+β 0
∂ψ ∗
∂x ∗
=
1
ρ 0 H 1
∇.(T ∗ ∧ e 3 ),
(9.36)
where
λ 0 =
f 2
0
g′
(
1
H 1
+
1
H 2
)=
1
L 2
D1
,
in the baroclinic response and λ 0 =0in the barotropic case. Note that from
section 5.3, the adjustment problem for the constant density case satisfies also
an equation of the form (9.34) but then with Λ=F (and λ 0 = f 2
0 /(gD)). Often a linear damping (−ε 0 ∇ 2
∗ ψ ∗ ) is added to the right hand side with a damping
coefficient ε 0 (s −1 ).
The equation (9.34) provides insight into the response of the flow to different
frequencies in the wind forcing. Consider the baroclinic case with H 2 ≫ H 1 and
hence F 2 ≪ F 1 . We can write
Λ=F 1 =(
L
L D1
)
2
(9.37a)
L
βUτ p
=
1
β 0 Lτ p
= τ β /τ p
(9.37b)
L
βUτ p
Λ=
L
β 0 L 2
D1 τ p
=
L
c r τ p
,
(9.37c)
where c r = β 0 L 2
D is the magnitude of the Rossby phase speed. The factor L/c r
is hence a wave time scale which is called the adjustment time scale τ c .
Ex. 9.3
With τ p being the time scale of variation of the wind forcing, the following
cases can be distinguished
(i) High frequency forcing, i.e., τ β /τ p ≫ 1 and τ c /τ p ≫ 1. In this case, fluctuations in the wind will not generate Rossby waves and the response is local.
(ii) Low frequency forcing, i.e., τ β /τ p ≪ 1 and τ c /τ p ≪ 1. In this case, the first
term in (9.34) can be neglected and the ocean is always in Sverdrup balance
with the changing wind stress.
(iii) On time scales τ p on the order of τ c or/and τ β Rossby waves are an important
component of the response.