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.