Adjustment
207
where G is an arbitrary function. As short Rossby waves will only play a role near
the western boundary (through reflection), the total solution outside the western
boundary can be well approximated by
Φ(x, t)=Φ
I (t)+Φ
L (x, t)=−
t
˜
Λ
+ G(x +
t
˜
Λ
).
(9.44)
With this solution, we can only satisfy the eastern boundary condition, Φ=0at
x =1,forallt>0.T h i sg i v e s
−
t
˜
Λ
+ G(1 +
t
˜
Λ
)=0,
(9.45)
with G(η)=(η − 1)H(η − 1) as a solution. From (9.44) the solution follows as
t< ˜
Λ(1 − x):Φ ( x, t)=−
t
˜
Λ
,
(9.46a)
t> ˜
Λ(1 − x):Φ ( x, t)=x − 1.
(9.46b)
Plots of the time development of the solution for ˜
Λ=1are shown in Fig. 9.4.
The time t ∗ = τ p ˜
Λ(1 − x) is exactly the time a long Rossby waves takes to
travel between the east coast and the location x. As according to (9.37) ˜
Λ=
L/(c r τ p ), the time needed to adjust to the Sverdrup flow at location x ∗ is hence
approximately given by (L−x ∗ )/(β 0 L 2
D1 ). For example, for Rossby wave speeds
for the first baroclinic mode, about 5 cms −1 , the adjustment time is in the order
of years. For the constant density case, it is (L − x ∗ )/(β 0 R 2
D ) and in the order of
days.
To satisfy the boundary conditions at the western boundary, short Rossby waves
are needed. These satisfy the short-wave approximation (9.30), i.e.,
Φ
S
xxt +Φ
S
x =0,
(9.47)
with Φ=0at x =0and a matching condition with the flow outside the boundary
layer. We determine the solution through the Laplace transform technique; for
s>0,define
φ(s, x)=L(Φ(x, t)) =
∞
0
Φ(x, t)e
−st dt.
(9.48)
It then follows from (9.47), with Φ(s, 0) = 0,that
φ(s, x)=α 1 (s)(e
−x/s − 1),
(9.49)
where α 1 (s) is still undetermined. When the long Rossby waves have not reached
the western boundary region, the matching condition becomes
lim
x→∞
φ(s, x)=L(
−t
˜
Λ
)=−
1
˜
Λs 2
⇒ α 1 (s)=
1
˜
Λs 2
.
(9.50)