Geography Reference
In-Depth Information
Unlike the phase speed, which is always westward relative to the mean flow, the
zonal group velocity for a Rossby wave may be either eastward or westward relative
to the mean flow, depending on the ratio of the zonal and meridional wave numbers
(see Problem 7.20). Stationary Rossby modes (i.e., modes with c
0 ) have zonal
group velocities that are eastward relative to the ground. Synoptic-scale Rossby
waves also tend to have zonal group velocities that are eastward relative to the
ground. For synoptic waves, advection by the mean zonal wind is generally larger
than the Rossby phase speed so that the phase speed is also eastward relative to the
ground, but is slower than the zonal group velocity. As indicated in Fig. 7.4b, this
implies that new disturbances tend to develop downstream of existing disturbances,
which is an important consideration for forecasting.
It is possible to carry out a less restrictive analysis of free planetary waves using
the perturbation form of the full primitive equations. In that case the structure of
the free modes depends critically on the boundary conditions at the surface and
the upper boundary. The results of such an analysis are mathematically compli-
cated, but qualitatively yield waves with horizontal dispersion properties similar to
those in the shallow water model. It turns out that the free oscillations allowed in a
hydrostatic gravitationally stable atmosphere consist of eastward- and westward-
moving gravity waves that are slightly modified by the rotation of the earth, and
westward-moving Rossby waves that are slightly modified by gravitational stabil-
ity. These free oscillations are the normal modes of oscillation of the atmosphere.
As such, they are continually excited by the various forces acting on the atmo-
sphere. Planetary scale-free oscillations, although they can be detected by careful
observational studies, generally have rather weak amplitudes. Presumably this is
because the forcing is quite weak at the large phase speeds characteristic of most
such waves. An exception is the 16-day period zonal wave number 1 normal mode,
which can be quite strong in the winter stratosphere.
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7.7.2
Forced Topographic Rossby Waves
Although free propagating Rossby modes are only rather weakly excited in the
atmosphere, forced stationary Rossby modes are of primary importance for under-
standing the planetary scale circulation pattern. Such modes may be forced by
longitudinally dependent diabatic heating patterns or by flow over topography. Of
particular importance for the Northern Hemisphere extratropical circulation are
stationary Rossby modes forced by flow over the Rockies and the Himalayas. It is
just the topographic Rossby wave that was described qualitatively in the discussion
of streamline deflections in potential vorticity-conserving flows crossing mountain
ranges in Section 4.3.
As the simplest possible dynamical model of topographic Rossby waves, we
use the barotropic potential vorticity equation for a homogeneous fluid of variable
depth (4.26). We assume that the upper boundary is at a fixed height H , and the
