An important conservation property involving a linear
combination of squares of the perturbation functions can
be obtained by taking the scalar product of 
with (4.18), the product of 
with (4.19) and
with (4.20) and adding the
results. This yields
To within the constraints imposed by the linearization of the basic equations and boundary conditions, the various terms in (4.21) have the following physical interpretations:
is the kinetic energy density of the perturbed state above and beyond that of the basic state;
is the baroclinic or internal gravitational potential energy density relative to the rest state associated with vertical displacement of the surfaces of potential specific volume;
is the bulk elastic or compressional potential energy density above and beyond that of the basic states; and
is the flux of energy which redistributes the energy
spatially within the fluid.
A possible definition of the group velocity 
in a
collective sense is
where 
.
We postpone further discussion of this until we examine
the solutions of the linear problem for particular wave modes.
It is sufficient to say at this point that the upper bound
for 
is the sound speed.
If we integrate (4.21) over the whole fluid volume and employ the Gauss divergence theorem we obtain
where S is the closed surface enclosing the fluid volume V. Application of the Liebnitz theorem gives
On the solid impermeable boundary 
vanishes. At the sea surface is essentially
for small surface slope and
where 
is the sea surface domain.
Eq. (4.25) is required by mass conservation.
Since 
is always one sign we might expect that the
integral of 
over will also vanish or at least
be of third order compared with the last integral in (4.24).
To this approximation (which is consistent with preceding
approximations of the basic equations)
Now from (3.5) and (3.12) we have (for 
constant)
so
or
Since 
is regarded as constant (4.25) then takes
the form
where the integral on the RHS applies at the water's edge.
If 
or 
= 0 at the edge then the latter integral vanishes.
In this case (4.28) reduces to
where
is the total available potential energy and where
, 
, 
are the volume integrals of
, 
, 
and
being the barotropic or surface gravitational
potential energy and 
is the surface elastic energy,
both relative to that of the basic state.
Thus the sum ( 
) is the total energy
of the perturbations about the basic equilibrium state and this
is conserved to the extent that there exists no external forcing
and no internal dissipative processes (i.e., no viscous, diffusive
or conductive effects).
As a further interpretation of we note that (4.30b) is simply
an approximation of the quantity
where is the area of the disturbed corrugated free surface
and 
is its equilibrium leveled state ( 
).
In fact
where 
is the angular slope of the surface and
for small
slope. QED.
While all five forms of perturbation energy can exist
simultaneously, they can exist in pairs for "pure" wave modes
as we will see later. We identify some possible pure modes:
In contrast, for small scale turbulent vortex modes the energy
is entirely kinetic.
Also, it should be noted that while modes (a) to (d) have essentially
equal partitioning of kinetic and potential (or elastic) energy on
the average, the partitioning between kinetic and potential and
kinetic energy for Rossby modes depends strongly on wave length.
Potential energy, particularly , dominates at planetary
(very large) scale, while kinetic energy dominates at subcritical
scale (critical scale being defined as the scale for which
and 
are nearly equal for Rossby modes).