The appropriate energy equation associated with the Rossby wave solution derived in the preceeding sections must be based on (4.10) which we repeat here in the form
where 
denotes the horizontal gradient operator.
If we multiply by 
we get
which may be rewritten in the form
Now multiply by
where f is the constant f adopted in the definition (4.4)
[ 
],
then (4.22) becomes
where
and
The first term in E is kinetic energy per unit area, the
second is potential energy per unit area. THe term
is the horizontal flux of energy per unit width
associated with the Rossby waves.
There is no difficulty identifying the first term in the
above relation for E with the kinetic energy since the
flow is nearly geostrophic
[ 
, using
(2.1) and (4.1)]. The interpretation of the second term
as potential energy is facilitated by use of relation (2.8)
which shows
Or, using (2.1a), (2.2) and (1.6)
which is part surface energy and part internal
gravitational energy. For 
, the primary part
is due to the surface term, while for 
the
internal part dominates.
The interpretation of the terms in the energy flux is more
difficult. It is not smiply the vertical integral
of 
as in previous analyses. This is because
the starting point in the analysis is the quasi-geostrophic
vorticity equation (4.10) and not the more basic LTEs.
It is sufficient to say that this is consistent
with the previous findings that the mean over
a wave cycle is 
as we will show next.
First consider the mean E using (4.24a) and (4.11).
These give since 
also
From (4.24b) we get for the mean
However, using (4.12) gives
If we devide this by 
and compare the result with
(4.17), we confirm that 
for the Rossby waves.
The other important result is (4.27), which shows that the
energy of the Rossby waves is in general not equally partitioned
except when 
. For long
Rossby waves ( 
or
) the potential energy dominates; while
for short Rossby waves ( 
or 
) the kinetic energy dominates.
This is in contrast to the Poincare gravity modes for which
it can be shown that
.