Geography Reference
In-Depth Information
12.5.1
Vertically Propagating Kelvin Waves
For Kelvin waves the aforementioned perturbation equations can be simplified
considerably. Setting
v
ˆ
=
0 and eliminating
w between (12.36) and (12.37), we
ˆ
obtain
ik
ˆ
−
iν
u
ˆ
=−
(12.38)
∂
ˆ
βy
u
ˆ
=−
/∂y
(12.39)
ν
m
2
1/4H
2
ˆ
ukN
2
−
+
+ˆ
=
0
(12.40)
Equation (12.38) can be used to eliminate in (12.39) and (12.40). This yields
two independent equations that the field of
u must satisfy. The first of these deter-
ˆ
mines the meridional distribution of
u and is identical to (11.47). The second is
ˆ
simply the dispersion equation
c
2
m
2
1/4H
2
N
2
+
−
=
0
(12.41)
where, as in Section 11.4, c
2
(ν
2
/k
2
).
=
1/4H
2
, as is true for most observed stratospheric
Kelvin waves, (12.41) reduces to the dispersion relationship for internal gravity
waves (7.44) in the hydrostatic limit (
If we assume that m
2
). For waves in the stratosphere
that are forced by disturbances in the troposphere, the energy propagation (i.e.,
the group velocity) must have an upward component. Therefore, according to the
arguments of Section 7.4, the phase velocity must have a downward component.
We showed in Section 11.4 that Kelvin waves must propagate eastward (c > 0) if
they are to be trapped equatorially. However, eastward phase propagation requires
m<0 for downward phase propagation. Thus, the vertically propagating Kelvin
wave has phase lines that tilt eastward with height as shown in Fig. 12.12.
|
k
||
m
|
12.5.2
Vertically Propagating Rossby-Gravity Waves
For all other equatorial modes, (12.34)-(12.37) can be combined in a manner
exactly analogous to that described for the shallow water equations in Section 11.4.1.
The resulting meridional structure equation is identical to (11.38) if we again
assume that m
2
1/4H
2
and set
N
2
/m
2
gh
e
=
For the n
=
0 mode the dispersion relation (11.41) then implies that
Nν
−
2
(β
|
m
| =
+
νk)
(12.42)
