The divergence theorem states that the net transport through the sidewalls of an enclosed volume is equal to the total divergence of the flux inside. We can use this mathematical identity to build conservation equations for substances present in the ocean.
As an example, imagine an experiment to measure the advective flux (recall: that means concentration times velocity) of oxygen through Admiralty Inlet (where most of the water passes into or out of Puget Sound). You would likely make measurements of oxygen concentration and velocity along a vertical cross-section that reaches from the ocean surface to the bottom and that crosses from one side of the passage to the other. By measuring the total transport of oxygen across this surface, you could infer the net production (or uptake from the atmosphere) of oxygen in Puget Sound. If there is more oxygen leaving with the surface outflow than entering in the deep inflow, we know three things:
- The divergence of oxygen flux within Puget Sound must be positive,
- We can calculate the magnitude of the oxygen divergence from our transport measurements,
- Oxygen must be added to the water inside at the the same rate (by river sources, atmospheric exchange, or biological processes, for example) in order to compensate for the divergence; otherwise the average oxygen concentration inside Puget Sound will decrease.
We can also take our budget principles down to very small (aka “infinitesimal”) volumes and develop differential forms of the budget equations, as we will do in this section.