If we form the dot product of 
on Eq. (III-3) we obtain
the scalar equation
This relates the rate of increase of the kinetic energy associated
with the macroscopic relative motion to the work done
per unit time by gravity and the tide producing forces and the work
done per unit time by the pressure field and the work done per unit
time by the viscous stresses 
. The quantity
is equivalent to 
and is
consequently a scalar. In terms of the components of
we have
The deflecting force 
contributes nothing to Eq. (43) since it is directed normal
to and hence the dot product is zero. That is, the
deflecting forces can perform no work on the fluid system.
If we employ equation (I-1), the mechanical energy equation can be expressed in the form
The term 
represents the relative
kinetic energy per unit volume,
is the advective
transport of kinetic energy, and 
can be
interpreted as the non-advective transport of kinetic energy.
The latter quantity represents two physical processes: the
work done by pressure and the work done by the viscous stresses.
We can gain further insight with regard to the energy equation
by dealing with the integrated form of the equation. Consider
a finite volume of fluid whose boundaries move with the fluid
velocity . In this case the mass remains constant
and Eq. (I-3) may be employed in evaluating the volume integral
of the left term of Eq. (58). Furthermore, 
can be expressed
in terms of the geopotential function 
as defined in Art. 2.06
and described in Art. 2.07 and 2.08:
The field of gravity relative to the earth is sensibly invariant
with respect to time, i.e.
, so that
But, from Eq. (56) we have
Thus, Eq. (I-3) can be employed also in evaluating
.
Accordingly, the integral of Eq. (53) following a given mass of fluid M is
where 
is the total kinetic energy of the mass M associated
with the motion of M relative to the earth, and 
is the
total geopotential energy (or energy of position in the earth's field of
gravity) of the mass M. These quantities are defined by
The symbol 
represents the combined tide producing forces
.
Eq. (59) can be put in a more appropriate form by introducing the following transformations:
The rate of change of the total energy
can then be expressed as
where 
or 
is the outward unit normal at the point
x, y, z on the surface. The physical interpretations of the
integrals on the right hand side are as follows:
represents the generation of mechanical energy per unit time within the volume by the work of tide producing forces;
represents the generation of mechanical energy per unit time within the volume by the work due to p associated with changes of the specific volume;
represents the dissipation of mechanical energy per unit time within the volume by the action of viscous stresses (which, as will be shown later, is always positive and hence always leads to dissipation of energy);
represents the generation of mechanical energy per unit time by the work of pressure at the boundaries of the volume associated with motion of the boundary relative to the earth; and
represents the generation of mechanical energy per unit time by the work of viscous stresses at the boundaries (which, unlike D, can be positive or negative. In terms of the above symbols Eq. (III-4) can be written
All of the above terms with the exception of D can be
positive or negative. The term 
is a general form
of which the special case where p is uniform is familiar.
In this case becomes simply 
which will be
recognized as the rate of work done by p in expanding
the volume V. However, if p is uniform within the volume
then, because p must be continuous across the boundaries,
will reduce to 
, and hence
will be zero if p is uniform. This
result is important because it means that the pressure within
or at the boundaries of the volume can lead to a production
of kinetic plus potential energy only if p is not uniform.
This, of course, is the usual situation.
The generation term can be expressed in terms of three
separate phenomenon. These are bulk compression associated
with pressure changes, changes in density due to diffusion of
salt, and changes in density due to non-advective flux of
thermal energy. The first of these represents rates of change
of intrinsic (or bulk elastic) energy and is important in
fluid acoustics. The second and third phenomenon represent
generation of energy by thermohaline processes and are presumed
to be of secondary importance compared with surface processes
in the case of the ocean. In the atmosphere, however, the
generation of mechanical energy by thermal processes is all
important. A detailed discussion of the various contributions
to the term will be deferred until a later chapter.
So far as the surface waves and the major ocean current systems
are concerned and are the major sources of
potential and kinetic energy. The term is, of course,
the sole generation term so far as tides are concerned.
Ordinary gravity waves, i.e. sea and swell, and capillary waves
of periods ranging from a fraction of a second to several minutes
derive their energy from both differential surface pressures
and tangential surface stresses. Thus, and 
are equally important in the generation of ordinary waves.
Actually the supply of energy through the term is split
up between waves and the major current systems of the oceans. This
source of energy for the currents represents the primary
agent which maintains the quasi-steady currents in the presence
of internal dissipation of energy by the viscous stresses.
If we regard the volume V to which the integrals of
Eq. (III-4) apply as representing that of the combined oceans
of the world, then and apply to the net effects
over the sea surface and the ocean bottom and lateral
boundaries. The major contributions to is due to
differential presures coupled with motions at the ocean-atmosphere
interface, since at the bottom and sies of the ocean
is zero. There are two
exceptions: (1) at the mouths of rivers where
does not vanish and energy is
added to the ocean and (2) on the rare occasions when the
ocean bottom is disturbed by seismic tremors or submarine volcanic
eruptions and energy is added to the ocean which may be observed
in the form of long period tsunami waves at the surface.
The term may logically be split into two parts:
The first of these surface integrals represents the supply of energy to waves and currents essentially by the wind stress at the sea surface. The second integral represents the supply of energy by bottom stresses. The contributions to the latter quantity, however, are predominantly negative and the net effect is a positive dissipation of energy rather than a positive supply. The negative of the second integral represents physically the work done by the water in moving over the bottom in the presence of resisting tangential stresses. This implies that there is slippage of the fluid along a solid surface. It is generally accepted that the fluid particles immediately adjacent to a solid surface adhere to the latter so that if the surface is stationary, then the fluid velocity is zero at the surface. However, in a thin laminar layer along the surface the velocity gradient can be quite extreme so that at the outer limit of the laminar layer (only a few millimeters removed from the boundary) the fluid velocity can attain a significant magnitude. For all practical purposes one can evaluate the surface viscous dissipation in terms of the velocity and stress at the effective boundary of the laminar skin of fluid adjacent to the boundary. Thus, in a sense, the second integral of Eq. (65) replaces effectively that portion of the volume dissipation D which applies to the laminar boundary layer.
In any event, the boundary layer dissipation is important only in the shallow seas, continental shelves, and tidal channels which border the ocean proper. It is only in these regions that the currents are of significant proportions near the bottom.
It is suggested, in this connection, that the volume dissipative
process D is responsible for the majority of dissipation of
energy in the ocean proper. In bays and estuaries, on the other
hand, it is suggested that at the bottom and sides
of these basins is responsible for the majority of energy
dissipation. This view is supported by the fact that the
effective hydraulic radius
in the case of the ocean is one fourth of the effective horizontal
diameter, while in the case of an estuary or bay the hydraulic
radius is approximately equal to the depth. Thus, for the North
Atlantic the effective hydraulic radius is of the order
of 800 miles while in the case of Chesapeake Bay or the
continental shelf of the Gulf of Mexico the effective hydraulic
radius is of the order of 40 feet. The ratio is about
.
It is possible that a significant portion of the energy of the ocean currents is dispersed laterally towards the shelf regions by advection associated with eddies in the main streams. This dispersed energy can then be dissipated on the continental shelves more effectively than in the ocean proper. This process has been suggested by Rossby and others, but the empirical evidence is not conclusive. Ekman's classical theory of wind drift currents, on the other hand, calls for downward dissipation of energy associated with vertical eddies with eventual dissipation by viscous stresses at the smallest scale of turbulent eddies within the upper 300 meters of the ocean.
In any event, whether the major share of the kinetic energy of the large scale streams is dispersed downward or laterally, it is in all events dispersed from large scale organized motion to small scale disorganized motion and is eventually dissipated at the smallest scale by viscous processes. The loss of kinetic energy of macroscopic motion represents a gain of kinetic energy at the microscopic scale which is measurable only in terms of the thermal state of the fluid.
For the oceans as a whole, the total kinetic and potential energy is sensibly the same from year to year. This implies that on the average the dissipation of energy balances the supply of energy at the sea surface together with the generation within the ocean by tidal and thermal processes. In view of the above discussion, however, the supply at the surface is very nearly sufficient to maintain the currents and waves in the presence of dissipative processes. Thus, for all practical purposes
It must be borne in mind that this relation is valid only
for the average situation for the oceans as a whole.
Significant changes in and can occur in certain
areas of the sea and the general relation Eq. (64) must be employed
in these situations.