Geography Reference
In-Depth Information
Thus, for example, westerly flow over an isolated mountain that primarily excites
a response at a given k will produce stationary waves with both positive and negative
l satisfying (10.65). As remarked in Section 7.7.1, although Rossby wave phase
propagation relative to the mean wind is always westward, this is not true of the
group velocity. From (7.91) we readily find that the x and y components of group
velocity are
k
2
l
2
∂ν
∂k
=
−
c
gx
=
u
+
β
(10.66)
k
2
l
2
2
+
∂ν
∂l
=
2βkl
k
2
c
gy
=
(10.67)
l
2
2
+
For stationary waves, these may be expressed alternatively with the aid of (10.65) as
2uk
2
k
2
2ukl
k
2
c
gx
=
l
2
,
c
gy
=
l
2
(10.68)
+
+
The group velocity vector for stationary Rossby waves is perpendicular to the
wave crests. It always has an eastward zonal component and a northward or south-
ward meridional component depending on whether l is positive or negative. The
magnitude is given by
c
g
=
2u cos α (10.69)
(see Problem 10.9). Here, as shown in Fig. 10.14 for the case of positive l, α is the
angle between lines of constant phase and the y axis.
Because energy propagates at the group velocity, (10.68) indicates that the sta-
tionary wave response to a localized topographic feature should consist of two
wave trains, one with l>0 extending eastward and northward and the other with
l<0 extending eastward and southward. An example computed using spherical
geometry is given in Fig. 10.15. Although the positions of individual troughs and
ridges remain fixed for stationary waves, the wave trains in this example do not
decay in time, as the effects of dissipation are counteracted by energy propagation
from the source at the Rossby wave group velocity.
For the climatological stationary wave distribution in the atmosphere the exci-
tation comes from a number of sources, both topographic and thermal, distributed
around the globe. Thus, it is not easy to trace out distinct paths of wave propa-
gation. Nevertheless, detailed calculations using spherical geometry suggest that
two-dimensional barotropic Rossby wave propagation provides a reasonable first
approximation for the observed departure of the extratropical time-mean flow from
zonal symmetry.
Rossby waves excited by isolated orographic features also play a significant role
in the momentum budget. Letting the amplitude coefficient be real in (7.90), the
