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.