Open Ocean 1-D Advection-Diffusion Models

File last modified 9 November 1998

17.4 Stable, Non-conservative Tracers: Solving for J/w

Remember our long term goal here: we first solve for K/w, then figure out what J is, then ultimately deal with radiocarbon, to get the clock information. The next step in our quest is to deal with non-conservative tracers. We will use oxygen (since it is well measured, and then relate it to CO₂ by stoichiometry. The general equation for a stable, non-conservative tracer is

Now the general form of the solution to this equation is

You can prove this by substituting the function back into the equation. The form of this equation is not difficult to understand. You have the first part being the conservative advective-diffusive functionality, and you've added to it a linear non-conservative portion. And yes, that's right Claudia, there are two constants to be determined by requiring that the function satisfies two boundary conditions. Now we go through the process of matching up the boundary conditions: i.e. make two equations for C(0)=C₀ and C(z_m)=C_m. Notice that when we do this, the J/w part creeps into the constants, since they need to be adjusted to make ends meet in the presence of consumption/production. Then we can express this function in terms of C₀ and C_m as

Comparing this to the stable-conservative equation, you see the added term on the RHS associated with consumption/production. It doesn't look exactly like the form we showed in the previous equation, but if you expand it out, it does reduce to the simpler form.

OK, let's look at the example station. Below is a plot of oxygen in this subrange. Along with the data (the red dots) are several curves, corresponding to different values of J^* (J/w). The dark green corresponds to negative J (consumption) while the dark blue corresponds to positive J (production). Notice that the effect of J^* is to enhance the curvature when it is negative (i.e. when there is consumption) and to reduce the curvature when it is positive. If, on the other hand, there was downwelling, then the opposite would be true. Further, if the boundary conditions were such that we had C₀ > C_m, the effects would be reversed again.

Now applying our "chi by eye", we would argue for an optimal value of J^*(O₂) approximately -0.018 (again, you can do much better with non-linear least squares).

GoTo Next Section
GoTo Lecture TOC
GoTo 12.747 TOC