Geography Reference
In-Depth Information
0 so that the basic state thermal wind
vanishes and the mean flow is barotropic. The phase speeds in this case are
As the first special case we let U
T
=
βk
−
2
c
1
=
U
m
−
(8.22)
and
β
k
2
2λ
2
−
1
c
2
=
U
m
−
+
(8.23)
These are real quantities that correspond to the free (normal mode) oscillations
for the two-level model with a barotropic basic state current. The phase speed
c
1
is simply the dispersion relationship for a barotropic Rossby wave with no y
dependence (see Section 7.7). Substituting from (8.22) for c in (8.18) and (8.19)
we see that in this case B
=
0 so that the perturbation is barotropic in structure. The
expression of (8.23), however, may be interpreted as the phase speed for an internal
baroclinic Rossby wave. Note that c
2
is a dispersion relationship analogous to the
Rossby wave speed for a homogeneous ocean with a free surface, which was given
in Problem 7.16. However, in the two-level model, the factor 2λ
2
appears in the
denominator in place of the f
0
/gH for the oceanic case. In each of these cases
there is vertical motion associated with the Rossby wave so that static stability
modifies the wave speed. It is left as a problem for the reader to show that if c
2
is
substituted into (8.18) and (8.19), the resulting fields of ψ
1
and ψ
3
are 180
◦
out of
phase so that the perturbation is baroclinic, although the basic state is barotropic.
Furthermore, the ω
2
field is 1/4 cycle out of phase with the 250-hPa geopotential
field, with the maximum upward motion occurring west of the 250-hPa trough.
This vertical motion pattern may be understood if we note that c
2
- U
m
<0,
so that the disturbance pattern moves westward
relative to the mean wind
.Now,
viewed in a coordinate system moving with the mean wind the vorticity changes
are due only to the planetary vorticity advection and the convergence terms, while
the thickness changes must be caused solely by the adiabatic heating or cooling
due to vertical motion. Hence, there must be rising motion west of the 250-hPa
trough in order to produce the thickness changes required by the westward motion
of the system.
Comparing (8.22) and (8.23) we see that the phase speed of the baroclinic
mode is generally much less than that of the barotropic mode since for average
midlatitude tropospheric conditions λ
2
10
−
12
m
−
2
, which is approximately
equal to k
2
for zonal wavelength of 4300 km.
1
As the second special case, we assume that β
≈
×
2
0. This case corresponds, for
example, to a laboratory situation in which the fluid is bounded above and below by
=
1
The presence of the free internal Rossby wave should actually be regarded as a weakness of the
two-level model. Lindzen et al. (1968) have shown that this mode does not correspond to any free
oscillation of the real atmosphere. Rather, it is a spurious mode resulting from the use of the upper
boundary condition ω =0atp = 0, which formally turns out to be equivalent to putting a lid at the top
of the atmosphere.
