Open Ocean 1-D Advection-Diffusion Models
File last modified 9 November 1998
17.1 Rationale
The main purpose of studying one-dimensional, open-ocean advection-diffusion models is pedagogical. It's not really considered "state-of-the-art", having been extensively exploited in the 1960s, and actually developed many decades before that. In fact, there are few parts of the ocean that can be regarded as even remotely satisfying the assumptions and requirements of this class of model, and even then it is highly debatable how generalizable the results of such a model really are. However, it is an interesting stage in the evolution of open ocean modeling, and has much to offer as a learning tool for understanding the process of ocean modeling (at least one approach, anyway).
We will focus our discussion on a paper Abyssal Carbon and Radiocarbon in the Pacific, by H. Craig (1969: J. Geophys. Res. 74, 5491-5506. The paper is not particularly easy to read, but it is a rather complete and in depth discussion of one dimensional advection-diffusion equations for a variety of different tracers in the deep Pacific. It is a refinement and correction of an earlier paper called Abyssal Recipes by W. Munk (1966: Deep-Sea Res. 13, 707-730) which seems to be the first to attempt to use 14C in a 1-D model to estimate upwelling rates and vertical turbulent diffusivity in the deep waters.
17.2 The General Setting and Equation
The basic situation is thus: there is a depth range in the deep Pacific between the very deep core of incoming "Common Water" (a mixture of North Atlantic Deep Water, Circumpolar Waters and Antarctic Bottom Water, which enters the Pacific at around 3500 - 4000 m depth) and the low salinity core associated with Antarctic Intermediate water at around 1000 m depth. Water properties are maintained at the ends of this range by horizontally flowing water, which fix the concentrations of the various properties. In between, the property concentrations are determined by a combination of vertical advection, vertical (turbulent) diffusion, in situ processes of biogeochemical production/consumption and radioactive decay. This depth range is referred to as an advective-diffusive subrange. Throughout this range, we would argue that the processes affecting tracer distributions are purely vertical, and hence subject to one-dimensional modeling.
Not all tracers exhibit the same behavior, as they are controlled by different physical and biogeochemical processes. One can break the tracers down into the following classes:
We will consider only passive tracers here (i.e. treat temperature and salinity as passive). The general equation for a generic steady-state, non-conservative, radioactive tracer in one dimension is
where z is the depth (positive upward, =0 at the bottom, zm
at the top), K is the vertical turbulent diffusivity (assumed
to be constant with depth), w is the vertical velocity (positive
upward), J is the in situ production rate (negative for
consumption), and 
is the radioactive
decay constant, but in general may represent some first order
(concentration dependent) consumption term. We will only use it
here as a radio-decay constant. A conservative tracer would have
J=0, and a stable, non-conservative tracer would have 
=0,
etc.
The objective is to determine the rates of upwelling and diffusion
in the deep Pacific, and to determine the rate of oxygen consumption/nutrient
remineralization. If you look at the equation, you can imagine
that the only constant really "known" in the equation
is the 
so that we will have to use some
radioactive tracer (radiocarbon is the one) to then determine
the other terms. The problem is that radiocarbon is non-conservative
(it has a non-zero J), so we really need to develop three equations
for the three unknowns. Thus we need to use three tracers.
Our general approach, as taken in the Craig paper, will be to use T and S to solve for the ratio of K/w, use oxygen to solve for the ratio of J/w (and then to infer the carbon "J") and to then use the radiocarbon distribution to solve for w. We can then work backwards to get J and K.
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