Open Ocean 1-D Advection-Diffusion Models

File last modified 9 November 1998

17.3 Stable, Conservative Tracers: Solving for K/w

The first step is to constrain K/w. First, let's think about what is happening in the deep Pacific. On the global scale, we have waters which are "formed" by density modification in the polar/subpolar oceans. This water is made very dense by extraction of heat (colder water is denser than warmer), and sinks into the abyss. North Atlantic Deep Water is formed predominantly in the Norwegian and Greenland Seas, and overflows the sills between Iceland and Greenland and between Iceland and Scotland. This water flows southward, being modified by mixing with low salinity Labrador Sea Water and salty Mediterranean Outflow Water. NADW flows all the way to the Antarctic Circumpolar Current, where it is further modified and combined with Weddell Sea Bottom Water. It enters the deep Pacific along the western boundary, and ultimately upwells through the water column to return at shallower depths. About 20-40 x 10⁶ m³/s of water enters the Pacific, and if the area of the Pacific is about 2-3 x 10¹⁴ m², then the upwelling rate must be of order 10^-7 m/s (order 3 m/y). Not very fast, but because the deep Pacific is so old, very significant.

We begin by writing the one dimensional, steady-state advection-diffusion equation for a stable, conservative tracer:

This one you can solve in your head! It is just

where a₁ and a₂ are constants of integration, and z^* is the scale height defined by

You can convince yourself that z^* is indeed a length by dimensional analysis: K is in m²/s, and w is in m/s. The bigger z^* is, the flatter the exponential curve is, and the more it looks like a straight line. The smaller z^* is, the more "bowed" the curve becomes.

Now what are a₁ and a₂? Mathematically, they arise from the solution, and must be set so that the function C becomes the observed concentrations at either end of the subrange. That is, they arise from the requirement of satisfying the boundary conditions of the problem. Put another way, the form of the function (i.e. the fact that it is an exponential) comes from the form of the differential equation. The constants of the function come from the boundary condition.This makes sense when you think about it.

Now we have the fact that C = a₁+a₂ when z=0 (the bottom of the subrange), and we have that C=a₁ + a₂exp(-z_m/z^*) when z=z_m (z_m is the height of the top of the subrange). Thus we can solve for a₁ and a₂ in terms of C₀ (the concentration at the bottom) and C_m (the concentration at the top). This gives the form that Craig reports:

Which is not too bad. We leave it as an exercise for the student to see how that form arose from the previous constraint. (Give it a try, it isn't hard at all!)

Let's get a feeling for this business. If the subrange is purely diffusive (i.e. w is 0), then the z^* becomes infinite, and the exponentials tend to 1. It is a little hard to see with Craig's function "f(z)" since you have a funny ratio: as z^* tends to infinity, the exponents become small, so we can expand the exponentials as infinite series, so that we have for large z^*

which degenerates the solution to a linear equation. Another way of seeing this is to look at the original equation, and you'll see that you have

whose solution is simply

a straight line. Oh yes, and two constants again: no surprise. You can then solve for the constants in the same way you did for the advective-diffusive case. So now we have, when w = 0 (or the scale height z^* is infinite) a straight line between the two end-members. As we "turn up" the vertical velocity, z^* becomes smaller, and the line between the two end-members becomes increasingly curved.

Lets look at some actual data (taken from WOCE line P6 at 32 S, 160 W). Below are profiles of T and S as well as a T-S relationship. You can demonstrate from the above equations that two stable conservative tracers must be linear functions of one another over an advection diffusion subrange. So looking at the T-S relationship, we look for a linear region between the salinity minimum of the Antarctic Intermediate Water (at about 1000 m) and the incoming Common Water (at about 3500 m).

The T-S relationship on the RHS is a blowup of the one on the left, and shows the mostly "linear" region between the salinity minimum of the AAIW and the deeper salinity maximum. The fact that the T-S relationship is not perfectly linear should remind us that the underlying assumptions of the model are not perfectly met. The end-member depths we choose are 1200 and 3300 m. So now, we do a little "chi by eye" (something we encourage you not to do in practice), and choose a z* which best fits the data. You can do better than this, by using non-linear least squares fitting of the functions above to the data.

Included in the plot are a variety of different curves corresponding to different z^* values. Note that a negative z^* means that you have downwelling (see the shape of the curve is now concave upward).

It seems that a z^* of 550 m is the best compromise. Note that the salinity distribution favors a z^* of around 500m, while the temperature distribution favors one of about 600m. The difference is significant, and related to the fact that the model is not realistic, but it is remarkable how well the model does do.

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