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 perhaps impossible to measure accurately.

We can, however, calculate the net evaporation knowing the salinity of the flow in \((S_{i})\) and out \((S_{o})\), together with a rough estimate of the volume of water \(V_{o}\) flowing out, where \(V_{o}\) is a volume flow in units of m³/s (figure \(\PageIndex{1}\)).

Figure \(\PageIndex{1}\): Schematic diagram of flow into and out of a basin. Values from Bryden and Kinder (1991).

The mass flowing out is, by definition, \(\rho_{o} V_{o}\). If the volume of the sea does not change, conservation of mass requires: \[\rho_{i} V_{i} = \rho_{o} V_{o} \nonumber \]

where \(\rho_{i}\), \(\rho_{o}\) are the densities of the water flowing in and out. We can usually assume, with little error, that \(\rho_{i} = \rho_{o}\).

If there is precipitation \(P\) and evaporation \(E\) at the surface of the basin and river inflow \(R\), conservation of mass becomes: \[V_{i} + R + P = V_{o} + E \nonumber \]

Solving for \(\left(V_{o} - V_{i}\right)\): \[V_{o} - V_{i} = (R + P) - E \nonumber \]

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

Because salt is not deposited or removed from the sea, conservation of salt requires: \[\rho_{i} V_{i} S_{i} = \rho_{o} V_{o} S_{o} \nonumber \] where \(\rho_{i}, S_{i}\) are the density and salinity of the incoming water, and \(\rho_{o}, S_{o}\) are density and salinity of the outflow. With little error, we can again assume that \(\rho_{i} = \rho_{o}\).

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 \(\PageIndex{1}\), solving \((\PageIndex{4})\) for \(V_{i}\) assuming that \(\rho_{i} = \rho_{o}\), and using the estimated value of \(V_{o}\), gives \(V_{i} = 0.836 \ \text{Sv} = 0.836 \times 10^{6} \ \text{m}^{3}/\text{s}\), where \(\text{Sv} = \text{Sverdrup} = 10^{6} \ \text{m}^{3}/\text{s}\) is the unit of volume transport used in oceanography. Using \(V_{i}\) and \(V_{o}\) in \((\PageIndex{3})\) gives \((R + P - E) = -4.6 \times 10^{4} \ \text{m}^{3}/\text{s}\).

Knowing \(V_{i}\), we can also calculate a minimum flushing time for replacing water in the sea by incoming water. The minimum flushing time \(T_{m}\) is the volume of the sea divided by the volume of incoming water. The Mediterranean has a volume of around \(4 \times 10^{6} \ \text{km}^{3}\). Converting \(0.836 \times 10^{6} \ \text{m}^{3}/\text{s}\) to \(\text{km}^{3}/\text{yr}\), we obtain \(2.64 \times 10^{4} \ \text{km}^{3}/\text{yr}\). Then, \(T_{m} = \left(4 \times 10^{6} \ \text{km}^{3}\right) / \left(2.64 \times 10^{4} \ \text{km}^{3}/\text{yr}\right) = 151 \ \text{yr}\). 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.