Chapter 7 - Some Mathematics:
The Equations of Motion

Chapter 7 Contents

7.4 Conservation of Mass and Salt

Conservation of mass and salt can be used to obtain very useful information about flows in the ocean. For example, suppose we wish to know the net loss of fresh water, evaporation minus precipitation, from the Mediterranean Sea. We could carefully calculate the latent heat flux over the surface, but there are probably too few ship reports for an accurate application of the bulk formula. Or we could carefully measure the mass of water flowing in and out of the sea through the Strait of Gibraltar; but the difference is small and it cannot be measured with sufficient accuracy (Bryden and Kinder, 1991).

We can, however, calculate the net evaporation knowing the salinity of the flow in Si and out So, together with a rough estimate of the volume of water Vo flowing out, where Vo is a volume flow in units of m3/s (Figure 7.1).

Figure 7.1 Schematic diagram of flow into and out of a basin. Values from Bryden and Kinder (1991).

The mass flowing in is, by definition, ρo Vo. If the volume of the sea does not change, conservation of mass requires:

ρi Vi = ρo Vo
(7.1)

where, ρi , ρo are the densities of the water flowing in and out. We can usually assume, with little error, that ρi = ρo.

If there is precipitation P and evaporation E at the surface of the basin and river in flow R, conservation of mass becomes:

Vi + R + P = Vo + E
(7.2)

Solving for (Vo - Vi ):

Vo - Vi = ( R + P) - E
(7.3)

which states that the net flow of water into the basin must balance precipitation plus river in flow minus evaporation when averaged over a sufficiently long time.

Because salt is not deposited or removed from the sea, and because the salinity of the Mediterranean has not changed, conservation of salt requires:

ρi Vi Si = ρo Vo So
(7.4)

Where ρi, Si are the density and salinity of the incoming water, and ρo, So are density and salinity of the out flow. With little error, we can again assume that ρi = ρ0.

An Example of Conservation of Mass and Salt
Using the values for the flow at the Strait of Gibraltar measured by Bryden and Kinder (1991) and shown in figure 7.1, solving (7.4) for Vi assuming that ρi = ρo, and using the estimated value of Vo gives Vi = 0.836 Sv = 0.836 × 10 6 m3/s, where Sv = Sverdrup = 106 m3/s is the unit of volume transport used in oceanography. Using Vi and Vo in (7.3) gives (R + P - E) = -4.6 × 104 m3/s.

Knowing Vi, we can also calculate a minimum flushing time for replacing water in the sea by incoming water. The minimum flushing time Tm is the volume of the sea divided by the volume of incoming water. The Mediterranean has a volume of around 4 106 km3. Converting 0.836 106 m3/s to km3/yr we obtain 2.64 104 km3/yr. Then, Tm = (4 106 km3)/(2.64 10 4 km3/yr) = 151 yr. The actual time depends on mixing within the sea. If the waters are well mixed, the flushing time is close to the minimum time, if they are not well mixed, the flushing time is longer.

Our example of flow into and out of the Mediterranean Sea is an example of a box model. A box model replaces large systems, such as the Mediterranean Sea, with boxes. Fluids or chemicals or organisms can move between boxes, and conservation equations are used to constrain the interactions within systems.

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