File last modified 1 December 1998
22.6 Adding Chemistry
At this point it might be useful to see how geochemistry comes into this section inversion business. So far all we have talked about is conservation of water mass, with some discussion of "other properties". But if you were to actually try to use geochemical tracers in a section inversion, where would that information show up in the equations? A very instructive paper is the one by Rintoul and Wunsch (1991) wherein they applied the section inversion technique to a sizable number of properties (mass, heat, oxygen and nutrients) to get their fluxes and budgets for the North Atlantic.
The best place to start is with Wunsch (1978), he wrote equation 22.3.4 as:
where:
subject to:
where po is the level of no motion, also an unknown. And just for completeness sake we also write down here:
depending on your sign convention and for any station pair j, the distance between them is given by:
In Rintoul and Wunsch (1991) they wrote down as their conservation equation:
where the terms:
account for the net cross isopycnal flux, for the moment we'll ignore these terms. That leaves us with the following for a conservation equation:
so if we allow dp to be approximated by 
and do some rearranging we get can turn Wunsch
(1978) equation into:
which looks a lot like Rintoul and Wunsch's 1991 equation, minus the 
(NOTE: vj(p)=vrel).
Essentially all one needs to do (well not all) to add chemistry is to include a
in the
appropriate place in the conservation equation.
In Wunsch (1978) the conservation equation is expanded and the separate terms are assigned to the appropriate part of the matrix equation:
where the first group are assigned to 
and the second group is assigned to Ab, giving rise to:
In Rintoul and Wunsch (1991) the expansion looks like:
in this case the first group is assigned to y and the second to Ex, giving rise to:
depending on your sign convention. I should think that by now the similarity between equations 22.6.7 and 22.6.9 would be pretty obvious. Note also that the chemistry variable is in both groups of equation 22.6.8.
GoTo Next Lecture