Stratification
187
Rossby waves with a constant LD.
In a flat bottom continuously stratified liquid layer characterized by a constant internal Rossby deformation radius L D , the dimensional dispersion
relation for Rossby waves is
σ ∗ = −
β 0 k ∗
n 2 π 2
L 2
D
+ k 2
∗ + l 2
∗
,n =0, 1,...
Here we have assumed that the external deformation radius R D is much
larger than L D and furthermore the quasi-geostrophic approximation has
been applied. The Rossby wave mode with n =0is the barotropic mode
and it has no vertical structure; the modes with n>0 are baroclinic
Rossby modes.
8.4.2. Properties of Rossby waves
The phase speed C of the waves with dispersion relation (8.43) is
C =
C x
C y
=
σ
k
σ
l
=
−
β
χ+k 2 +l 2
−
β
χ+k 2 +l 2
k
l
,
(8.45)
such that the phase speed in x direction is always negative. An observer moving
with the waves always sees a larger planetary vorticity to the right.
The maximum frequency σ of Rossby waves is found for k =(l 2 + χ) 1/2 with
amplitude
σ m = −
βk
2(l 2 + χ) 1/2 .
(8.46)
The absolute maximum σ M occurs for l =0with
σ M = −
β
2χ 1/2 .
(8.47)
For l =0 , the frequency σ is plotted as a function of k/
√ χ in Fig. 8.6a. The
dimensional wavelength and frequency of this wave are
λ ∗ =
2πL
k
=
2πL
√ χ
,
(8.48a)
σ M ∗ =
σ M U
L
= −
βU
2
√
χL
.
(8.48b)
For constant S, with eigenvalues χ given by (8.40), the Rossby waves with largest
frequency have wavelengths
λ n∗ =
2πL
n
√
S =
2
n
L D ,
(8.49)
187
Rossby waves with a constant LD.
In a flat bottom continuously stratified liquid layer characterized by a constant internal Rossby deformation radius L D , the dimensional dispersion
relation for Rossby waves is
σ ∗ = −
β 0 k ∗
n 2 π 2
L 2
D
+ k 2
∗ + l 2
∗
,n =0, 1,...
Here we have assumed that the external deformation radius R D is much
larger than L D and furthermore the quasi-geostrophic approximation has
been applied. The Rossby wave mode with n =0is the barotropic mode
and it has no vertical structure; the modes with n>0 are baroclinic
Rossby modes.
8.4.2. Properties of Rossby waves
The phase speed C of the waves with dispersion relation (8.43) is
C =
C x
C y
=
σ
k
σ
l
=
−
β
χ+k 2 +l 2
−
β
χ+k 2 +l 2
k
l
,
(8.45)
such that the phase speed in x direction is always negative. An observer moving
with the waves always sees a larger planetary vorticity to the right.
The maximum frequency σ of Rossby waves is found for k =(l 2 + χ) 1/2 with
amplitude
σ m = −
βk
2(l 2 + χ) 1/2 .
(8.46)
The absolute maximum σ M occurs for l =0with
σ M = −
β
2χ 1/2 .
(8.47)
For l =0 , the frequency σ is plotted as a function of k/
√ χ in Fig. 8.6a. The
dimensional wavelength and frequency of this wave are
λ ∗ =
2πL
k
=
2πL
√ χ
,
(8.48a)
σ M ∗ =
σ M U
L
= −
βU
2
√
χL
.
(8.48b)
For constant S, with eigenvalues χ given by (8.40), the Rossby waves with largest
frequency have wavelengths
λ n∗ =
2πL
n
√
S =
2
n
L D ,
(8.49)