186
DYNAMICAL OCEANOGRAPHY
-1.5
-1
-0.5
0
0.5
1
1.5
-1
-0.8
-0.6
-0.4
-0.2
0
n = 0 (barotropic)
n = 1
n = 2
n = 3
Φ Φ
Φ
Φ(z)
z
(a)
N
2
(s )
-1
depth
S = constant
(b)
Figure 8.5. (a) Structure functions Φn(z) for the first four modes n =0 , 1, 2 and n =3for a
N
2 profile such as in (b).
are plotted in Fig. 8.5b. The density profile ¯
ρ that corresponds to a constant S
(Fig. 8.5a) is linear.
Through separation of variables, the function Ψ(x, y, t) in (8.36) is determined
from
∂
∂t
(∇
2 Ψ − χΨ) + β
∂Ψ
∂x
=0.
(8.41)
To determine free waves in a horizontally unbounded domain, we substitute traveling waves solutions of the form
Ψ(x, y, t)=Ψ 0 e
i(kx+ly−σt) ,
(8.42)
into (8.41), where k and l are the wavenumbers in the x-a n dy-direction, σ is
the frequency of the wave and Ψ 0 is its amplitude. For a fixed value of χ n ,t h e
dispersion relation follows as
σ n = −
βk
χ n + k 2 + l 2 .
(8.43)
Hence, for each value of χ n , there is an associated frequency σ n and because χ n is
real, the corresponding σ n are also real. For the barotropic mode we have χ 0 =0
and the frequency of these waves is
σ 0 = −
βk
k 2 + l 2 .
(8.44)