Stratification
189
where C g is the group velocity, with
C g =
∂σ
∂k
∂σ
∂l
.
(8.54)
For the Rossby waves (8.43), we find for the group velocity
C g =
β
k 2 −l 2 −χ
(k 2 +l 2 +χ) 2
β
2kl
(k 2 +l 2 +χ) 2
.
(8.55)
Waves for which C g = C are called dispersive waves and the Rossby wave is
hence an example of these type of waves. From the expression for C x
g , we can
deduce that a wave packet that contains a wave with k 2 >l 2 + χ will move
eastward while long Rossby waves with k 2 the absolute values of the group velocity, we find | C x
g |≤| C x |, as can be seen in
Fig. 8.6b for waves with l =0.
Additional Material
D: Properties of Rossby waves are discussed in all textbooks on geophysical
fluid dynamics, for example chapter 15 of Cushman-Roisin (1994) and chapter 4 of Mc Williams (2006).
D: Material for further study can be found in section 5.7 of Vallis (2006), sections
6.11, 6.12 and 6.15 of Pedlosky (1987) and lectures 14, 15 and 19 of Pedlosky
(2003).
8.4.3. Topographic Rossby waves
Bottom topography can substantially affect the propagation of Rossby waves.
We will study this using an example of a simple bottom profile given by
η b (y)=γy,
(8.56)
and assume that S is constant. The equation (8.29) with the boundary conditions
(8.33-8.34) becomes
∂
∂t
(∇
2 ψ +
1
S
∂ 2 ψ
∂z 2 )+β
∂ψ
∂x
=0 ,
(8.57a)
z = −1:
∂ 2 ψ
∂t∂z
= −γS
∂ψ
∂x
,
(8.57b)
z =0:
∂ 2 ψ
∂t∂z
=0 .
(8.57c)