Equatorial ocean circulation
257
which gives a bounded wave called the Yanai wave. For large σ, the character
of the wave becomes Kelvin like, whereas for small σ it becomes Rossby like. A
classical picture of the dispersion relation for the Kelvin wave, the Yanai wave and
j =1Rossby and inertia-gravity waves is plotted in Fig. 11.10. The Yanai and
0
0.5
1
1.5
2
2.5
3
-4
-3
-2
-1
0
1
2
3
4
Rossby, j = 1
k
inertia-gravity , j = 1
σ σ σ
σ
Yanai
Kelvin
Figure 11.10. Dispersion relation of equatorial free waves. Shown are the Kelvin wave, the Yanai
wave and the j =1Rossby wave.
Kelvin waves have a positive group velocity and for inertia-gravity and Rossby
waves, the group velocity c g becomes
c g =
∂σ
∂k
=
1+2σk
2σ 2 +
k
σ
.
(11.38)
For low frequency Rossby waves (Fig. 11.10), the group velocity is negative and
the approximate dispersion relation is (note that both k 2 << 1 and σ 2 << 1)
σ = −
k
2j +1
.
(11.39)
Their dimensional phase velocity is given by
c ∗ = −
c o
2j +1
,
and depends on the meridional wavenumber j. In the limit ζ o → 0, only these
long waves remain which can be immediately concluded from (11.26). This limit