Geography Reference
In-Depth Information
where as usual we have neglected terms involving the products of perturbation
quantities. We seek a solution of the form
Re
exp(iφ)
ψ
=
where φ
νt. Here k and l are wave numbers in the zonal and meridional
directions, respectively. Substituting for ψ
in (7.90) gives
=
kx
+
ly
−
ku)
l
2
k
2
(
−
ν
+
−
−
+
kβ
=
0
which may immediately be solved for ν:
βk/K
2
ν
=
uk
−
(7.91)
where K
2
k
2
l
2
≡
+
is the total horizontal wave number squared.
=
Recalling that c
ν/k, we find that the zonal phase speed relative to the mean
wind is
β/K
2
c
−
u
=−
(7.92)
which reduces to (7.88) when the mean wind vanishes and l
0. Thus, the
Rossby wave zonal phase propagation is always
westward
relative to the mean
zonal flow. Furthermore, the Rossby wave phase speed depends inversely on the
square of the horizontal wavenumber. Therefore, Rossby waves are dispersive
waves whose phase speeds increase rapidly with increasing wavelength.
This result is consistent with the discussion in Section 6.2.2 in which we showed
that the advection of planetary vorticity, which tends to make disturbances ret-
rogress, increasingly dominates over relative vorticity advection as the wavelength
of a disturbance increases. Equation (7.92) provides a quantitative measure of this
effect in cases where the disturbance is small enough in amplitude so that perturba-
tion theory is applicable. For a typical midlatitude synoptic-scale disturbance, with
similar meridional and zonal scales (l
→
k) and zonal wavelength of order 6000 km,
the Rossby wave speed relative to the zonal flow calculated from (7.92) is approx-
imately
≈
8ms
−
1
. Because the mean zonal wind is generally westerly and greater
than8ms
−
1
, synoptic-scale Rossby waves usually move eastward, but at a phase
speed relative to the ground that is somewhat less than the mean zonal wind speed.
For longer wavelengths the westward Rossby wave phase speed may be large
enough to balance the eastward advection by the mean zonal wind so that the
resulting disturbance is stationary relative to the surface of the earth. From (7.92)
it is clear that the free Rossby wave solution becomes stationary when
−
K
2
K
s
=
β/u
≡
(7.93)
The significance of this condition is discussed in the next subsection.
