128
6 Rotational Effects
where T is wave period, λ is the true wavelength, λ x is the apparent wavelength
measured along bathymetric contours, α is the bottom slope, and R is the Rossby
radius of deformation, given by (6.6). Consequently, the phase speed of wave propagation along topographic contours is given by:
c x =
λ x
T
=
αg
f
1 + (2π) 2 (R/λ)
2
(6.28)
which implies that topographic Rossby waves propagate with shallower water on
their right (left) in the northern (southern) hemisphere.
The following example gives an estimate of the phase speed of these waves.
The deformation radius is R = 100 km for a depth of 100 m at mid-latitudes
( f = 10
−4
s
−1
). Given a wavelength of λ = 10 km and a bottom slope of α = 0.01
(corresponding to bathymetric variation of 10 m over 1 km), the phase speed of topographic Rossby waves is about 0.25 m/s or 22 km per day.
Planetary Rossby waves in a flui of uniform density follow the dispersion relation (Cushman-Roisin, 1994):
T =
λ x
β R 2
1 + (2π )
2
(R/λ)
2
(6.29)
where λ x is the apparent wavelength measured in the zonal direction. Here, the zonal
phase speed of wave propagation is given by:
c x = −
β R
2
1 + (2π) 2 (R/λ)
2
(6.30)
This zonal phase speed is always negative, implying a phase propagation with
a westward component. The deformation radius is R = 2200 km for a deep-ocean
depth of 5000 m at mid latitudes ( f = 10
−4
s
−1
). With a wavelength of λ = 100 km
and β = 2.2 × 10
−11
m
−1
s
−1
, we yield a phase speed of 5.5 mm/s corresponding to
a distance of 175 km per year. Hence, planetary Rossby waves usually propagate at a
much slower speed compared with topographic Rossby waves found predominantly
at continental margins.
For relatively short waves, λ << R, the latter equation reduces to:
c x = −
βλ
2
(2π) 2
(6.31)
which implies that the zonal phase speed increases for larger wavelengths.