Adjustment
203
The angle of incidence is therefore equal to the angle of reflection (θ i = θ r ) and
the frequency is invariant under reflection. The wavenumber of the reflected wave
must therefore lie in the same circle as the incoming wave (see Fig. 9.2b).
Because the energy of the reflected wave must propagate eastward, the reflected
Rossby wave must have a short wavelength and hence the wavenumber is given
by
k r = −
β
2σ i
+
β 2
4σ 2
i
− (χ + l 2
i ),
(9.26)
and k r >k i . The latter follows from (9.24); A i = −A r , such that the amplitude is
the same but the phase is shifted by π. The conclusion is that long Rossby waves
are reflected as short Rossby waves.
For long Rossby waves with l =0the dispersion relation in the limit k → 0
becomes (see 9.19)
σ = −
βk
χ
.
(9.27)
These waves satisfy the long-wave approximation
β
∂ψ
∂x
− χ
∂ψ
∂t
=0.
(9.28)
For short waves (k →∞in (9.19)) the dispersion relation is
σ =
−β
k
,
(9.29)
and these waves satisfy the short -wave approximation
∂
∂t
∂ 2 ψ
∂x 2 + β
∂ψ
∂x
=0.
(9.30)
We will use the approximate equations (9.28) and (9.30) to solve for the adjustment problem in the next section.
9.3. Adjustment in a rectangular basin
Consider a rectangular basin as in Fig. 9.3 (with length L, width B, and aspect
ratio d = B/L) having a flat bottom (η b =0 ) . The dimensionless wind stress T
is given by
T(x, y, t)=f (t)
⎛
⎝
− cos(πy/d)
0
0
⎞
⎠ ,
(9.31)
where f (t) has not yet been specified.