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# The Continuity of Salt

Let represent the mass of a particular dissolved constituent of sea water per unitmass of water. The sum of the major dissolved constituents, by mass, per unit mass of sea water will by denoted by

The concentration, i.e. mass per unit volume of sea water, for the individual constituents is , and for the major constituents as a whole is .

We will include only those constituents in the sum S whose concentrations are no influenced to any significant extent by chemical or biological processes in the sea. Essentially, the quantity S represents the salinity of the sea water. The salinity is said to be conservative in the sense that it is dependent upon physical processes only.

Consider an arbitrary volume of water whose boundaries movie with the fluid velocity. The total mass of the volume as noted in article 4.03 must remain constant. However, the mass of the major dissolved constituents can change due to diffusion. If we let represent this mass at time t then

Let represent the diffusive flux of salt which exists at any point x,y,z i the water at the instant t, by virtue of a gradient in the salt concentration. The diffusive flux represents a non-advective (i.e. not related to the fluid velocity) transport of mass of salt through a unit area normal to the direction of the flux per unit time. The net amount of salt leaving the volume V per unit time due to diffusion across its boundaries can be expressed as

where is the outward unit normal to the surface A.

If we exclude the possibility of precipitation of salt within the volume due to supersaturation then diffusion along the

gradient represents the only mechanism by which can change with time for a volume moving with the fluid. Consequently

which, in view of the divergence theorem, can be written as

Now, from Eq. (14)

but in view of Eq. (I-3) or (12) we can express this as

Combining this with Eq. (16), and recalling that V is arbitrary, yields the following differential equation for the conservation of salt:

An alternate form can be obtained by adding this equation to Eq. (I-1) multiplied by S. The result is

Eq. (I-5) is somewhat analogous to Eq. (II-1) where the concentration takes the place of , but in this case the additional term is present. This suggests that we can write an equation similar to (II-2) for the conservation of water, and that the sum of these equations must be consistent with Eq. (I-1).

The concentration of water is equal to . Thus, if we let represent the diffusive flux of water in the sea water mixture, then it can be shown by the same line of reasonoing as that which leads to Eq. (II-2) that

If we add Eqs. (II-2) and (II-3) we obtain

But, in view of Eq. (I-1) it follows that and are not independent, but are related by

in order that the total mass be conserved. Eq. (20) can be satisfied throughout the fluid only if

To shed further light on this result it is instructive to introduce what might be termed the diffusion currents of salt and water, and . These are defined as follows:

and

These velocities cannot me measured directly but must be inferred from measurements of the diffusive fluxes, which are in turn deduced from measured changes in the concentration of a particular mass of fluid.

It follows from Eq. (21) that

The normal value of S in the ocean being about 35 ppt indicates the the diffusive current is only about 3.6% of in magnitude and is always in a direction opposite that of .

If we replace the terms and by the equivalent expressions in terms of and , Eqs. (II-2) and (II-3) take the form

The total velocity of the salt particles (in a statistical mean sense) is and that of the water particles is . Since and are always opposite in direction such as to yield a net total flux of mass of zero, the overall weighted mean velocity is therefore , the velocity of the ensemble of fluid particles as a unit. It is this velocity which is capable of being measured directly.

Next: Advective and Non-Advective Transport Up: The Fundamental Equations of Previous: The Equation of Continuity

Steve Baum
Mon Dec 1 08:50:29 CST 1997