Energy Flow and Energy Equations

Hydrodynamic Equations of the Sea

4.10 Energy Flow and Energy Equations

extant values of 0 and s. Since most of the water in the ocean is deep, the potential temperatures are low. The three major oceans have distinctive ther- modynamic properties, as Fig. 4.12 shows.

Thermodynamics and Energy Relations 153

A somewhat more specialized form of this can be obtained for cases where the potential energy is due to wavelike elevations,

t .

Using Eq. 3.120 for the kinematic boundary condition and assuming that the potential gradient has only a z component, we obtain

(4.69) We can then rewrite Eq. 4.68 with terms in the amplitude,

t ,

transformed to the left-hand side:

a

- at ( % pu‘

+

p g t )

+ v

* [ ( % pu’

+

^pgl

+

^{p )} u - A .

v

(’h u’)]

= p v * u ^- p c . (4.70)

This form of the energy equation is useful for wave propagation problems.

Returning to the more general case Eq. 4.68, define now a mechanical ener- gy flux vector, F,,, having dimensions of watts per square meter, by

F,, = (9” pu’

+

p ) u -

A.v(%

u‘)

.

(4.71) Then Eq. 4.68 becomes

a

at

- ( % p u 2 )

+

v . F , , = - p u . V @

+

^{P V . U - pc}

.

(4.72) This equation can be interpreted with the use of the parallelepiped of Fig.

4.13. First, the kinetic energy per unit volume within the parallelepiped is changing at a rate given by the time derivative on the left-hand side; the oth- er term on the left is the divergence of the flux of mechanical energy across the faces of the small volume, V .F,. On the right-hand side are (1) the rate of working by or against gravitational forces when fluid crosses equipoten- tial surfaces, - p u . V @ = - p w g (the latter form being valid when the earth’s gravity is the only source of potential energy); (2) the rate of energy release or uptake by fluid expansion or compression, p v . u = @ / a ) D a / D t ; and (3) the rate of loss of energy within the parallelepiped due to viscous forces.

This latter term needs interpretation, an attempt at which will be given below.

Let us return to the energy flux vector, F,,: Equation 4.71 shows it to be composed of the net flux rate of kinetic energy per unit volume, % pu2u,

across all six faces; the net energy flux due to pressure forces, pu; and the

154 Principles of Ocean Physics

t

Advective flux:

-

( 1 / 2 p u2 + p ) u Diffusive flux:

Fig. 4.13 Advective and diffusive fluxes of mechanical energy through an elemen- tal volume of water, plus dissipation and conversion within the volume, as given by Eq. 4.72. [After Gill, A. E., Atmosphere-Ocean Dynamics (1982).]

diffusive flux of kinetic energy across the surfaces caused by eddy and molecu- lar viscosities. As with all geophysical fluid eddy processes, this rate of ener- gy diffusion in the vertical is much smaller than the rate in the horizontal.

The energy dissipation rate, as written, includes only molecular viscosity.

The only true frictional dissipation in the system is due to the molecular forces and occurs at very small lengths, near

I,

= ( v , ~ 3 / ~ ) which is of milli- meter order. This dissipated mechanical energy is transformed into heat and, at short wavelengths, into potential energy of vertical mixing; these process- es contribute to the term Q ( c , 1 , L V ) in Eq. 4.40. Now the rich variety of flows in fluid dynamics makes it a practical impossibility to deal numerical- ly at the same time with scales ranging from the dimensions of an ocean basin (say lo4 km) to the scale of the dissipation length, I , , a span of the order of 10". At any given scale of study, the uses of an eddy viscosity, A, and its duals for heat and salinity diffusion, K~ and ^{K ~ ,} respectively, repre- sent an attempt to parameterize the so-called sub-grid eddy processes, as well as to remove energy from the model system, either analytically or numeri- cally, in a moderately realistic way. As we have seen, numerical values of A,, / p are of order lo2 to lo4 times molecular, while those for A h l p may be 10" to 10" times greater (of the order of very viscous, heavy syrup).

Nevertheless, it is only at the very small scales that conversion of mechani- cal energy to heat energy actually occurs; however, this heating rate, com- pared with heating from sunlight or cooling by evaporation, is essentially negligible. For example, a hydraulic flow that falls through a vertical drop of roughly 400 m and is completely randomized increases its temperature by only approximately 1°C.

Thermodynamics and Energy Relations 155

I t is a somewhat amusing fact that the initial energy inputs to the sea from wind stress and sunlight, as described in Chapters 2 and 9, and which occur at millimeter and submillimeter scales, are transformed by a wide variety of processes into global-scale flows, but then are ultimately dissipated by radi- ative and resistive processes again taking place at these small scales.

The budget of kinetic energy flux discussed in the context of Fig. 4.13 can be expanded to entire basin scales by considering the ocean to be composit- ed from many such elemental volumes. By integrating Eq. 4.72 over an ocean basin, for example, an equation for the rate of change of total mechanical energy is obtained. Let us define the total kinetic energy, K, to be

K = 54 pu2 d3x

.

(4.73)

Then applying the divergence theorem and the rule for differentiating under the integral to Eq. 4.73, we obtain

dK - = -

dr s F,, .fi d2x ^-

iji

v ^{P E d3x}

+ ![!

( - p u - V @

+ p v . 4

^{d 3 x ,} (4.74)

V

where (as before) fi is the unit outward normal to the boundary surface, i.e., bottom plus free surface. On the right-hand side are the expressions for the fluxes of energy into the bottom and across the air-sea interface, and the dissipation of kinetic energy within the volume. The final volume integral gives the rate at which the geopotential and the compressive forces do work;

these will be shown below to represent energy transformations from mechan- ical to other types.

There remain two other forms of energy to include in an overall energy rate equation: potential and internal. The geopotential, 9, is the potential energy per unit mass, since its negative gradient gives the combined gravita- tional and centrifugal forces per unit mass. Its astronomical tidal part is time- and space-dependent (cf. Chapters 3 and

3,

while the terrigeneous (earth- derived) portion depends only on z. An equation for its rate of change may be derived from the relationship for the convective derivative. The rate of doing work per unit volume, p D 9 / D t , is just the rate at which the geopoten- tial does work on the fluid flow, - p u . f n = p u . V 9 . From Eqs. 3.50 and 4.44 we obtain

(4.75)

This represents a budget. equation for potential energy.

from Eqs. 4.44, 4.54, 4.55, 4.58, and 4.59. Collectively, these become The rate equation for internal energy per unit mass, e, may be derived

a ( p e )

+ v .

^(peu)

p - = - De

Dt at

Ds

Dt

1

= p [ T $ - p - + p - + L Dol Dt

(4.76) or, rewriting this relationship slightly,

= Q

+

^{p S}

+

^L - ~ V * U . (4.77)

The quantity F, may be called the internal energy flux vector, and is given by

F, = peu

+

( S ) - ^K~ v T - ~ K ; V ( P S ) , (4.78) which shows that internal energy is transported by conduction, radiation, and eddy and molecular diffusive heat transfer caused by both temperature and salinity gradients. The salinity contribution is undoubtedly small but could be appreciable under limited circumstances such as freezing or thaw- ing of seawater. On the right-hand side are volumetric sources of heat, Q, chemical energy released by salinity changes, pS, heats of phase transforma- tion, L , and heat released or absorbed by volumetric changes, - p V .u.

Thermodynamics and Energy Relations