Taking x eastwards and y northwards, the Cartesian form of (4.3) is
As an approximation we treat both 
and 
as independent of y, an inconsistency which can cause
trouble at very low latitudes. This approximation is therefore
restricted to mid- to high-latitudes and is often called
the `` -plane'' approximation, simply to identify it.
WIth this approximation, plane wave solutions of the form
are possible where 
is the horizontal wave number
as before. In order to avoid confusion with the
parameter, the x, y components of
will now be denoted by 
and 
, respectively
(i.e., 
.
Substitution of (4.11) in (4.10) then yields the
dispersion relation for mode n
This relation yields a maximum value of 
given by
which occurs for 
, 
.
It is of interest to note the this maximum Rossby wave frequency
is independent of 
but depends on the earth's curvature
and gravity (via 
). Specifically
However, in order for the quasi-geostrophic vorticity equation
(4.10) to be valid, and hence
should be much less than the local
value of f. This is satisfied if
The most severe restriction is on the barotropic mode for
which 
and
for D = 4000 m.
This shows that the approximation certainly is not valid
for the equatorial belt 
.
In fact for that region one must employ the more exact
LTEs (2.10) to (2.12) with allowance for variable f,
to examine the gravity and Rossby wave
modes simultaneously (Moore and Philander, Ch. 8,
Vol. 6, The Sea, 1977).
With the above caveat regarding the latitudes for which quasi-geostrophic theory holds, we now examine some of the interesting implications of the Rossby dispersion relation (4.12).
The phase velocity is given by
The x-component of this is
and is westward for all , .
The y-component is
which can be north or south depending on the sign
of 
.
The group velocity 
has the components
and
,
given by
which can be eastward or westward, and
which has a sign opposite to the y-component of 
.
The x-component of vanishes for
; it is
westward for smaller 
and eastward for
larger .
For 
whose magnitude is the maximum Rossby wave speed. For the
barotropic Rossby waves ( 
) this maximum speed at
is of order 80 m/s. The maximum speed for
the baroclinic modes is at most about 0.04 m/s and decreases
with increasing n.