Free waves
169
Summary
In a constant density rotating (with constant f 0 ) shallow-water layer
of constant depth H 0 without horizontal boundaries there are only
Poincar´ e waves with a dispersion relation
σ = ±
f 2
0 + C 2
0 (k 2 + l 2 )
where C 0 =
√ gH 0 .
When meridional boundaries are present with a distance L apart,
Kelvin waves can occur. These waves have a typical length scale R D
and a frequency σ given by
R D =
C 0
f 0
; σ = ±C 0 k
When bottom topography and/or the β-effect is present, Rossby waves
can occur. In case of a linear slope H 0 (y)=D(1−sy/L), with s ≪ 1
these Rossby waves have a dispersion relation on the β-plane given by
σ = −(
sf 0
L
+ β 0 )
k
k 2 + l 2 + f 2 /C 2
0
In the quasi-geostrophic approximation, the Poincar´ e waves are filtered and only Rossby waves can be represented.
169
Summary
In a constant density rotating (with constant f 0 ) shallow-water layer
of constant depth H 0 without horizontal boundaries there are only
Poincar´ e waves with a dispersion relation
σ = ±
f 2
0 + C 2
0 (k 2 + l 2 )
where C 0 =
√ gH 0 .
When meridional boundaries are present with a distance L apart,
Kelvin waves can occur. These waves have a typical length scale R D
and a frequency σ given by
R D =
C 0
f 0
; σ = ±C 0 k
When bottom topography and/or the β-effect is present, Rossby waves
can occur. In case of a linear slope H 0 (y)=D(1−sy/L), with s ≪ 1
these Rossby waves have a dispersion relation on the β-plane given by
σ = −(
sf 0
L
+ β 0 )
k
k 2 + l 2 + f 2 /C 2
0
In the quasi-geostrophic approximation, the Poincar´ e waves are filtered and only Rossby waves can be represented.