Advection-Diffusion Equations and Turbulence
File last modified 2 November 1998
15.1 Rationale
Why should we deal with advection-diffusion equations? They represent, after all, physical processes, which are the domain of the physical oceanographers. The problem is that a significant part of chemical oceanography involves the interpretation of the distributions of geochemicals in the water column, and these distributions are the result of the combined effects of physical, biological, chemical and geological processes. These processes are intimately intertwined in a way which requires us to "solve the physics" before we can begin to learn anything about the chemistry. In fact, there is an active branch of geochemistry which involves the use of geochemical tracer distributions to make inferences about ocean circulation and mixing. That is, some people make a living from this kind of stuff. The processes involved play a fundamental role in the ocean's ability to exchange, sequester and transport heat, carbon dioxide and other biologically important properties. This in turn feeds back into the role of the oceans in regulating climate, global primary production and all sorts of socially relevant things.
Put another way, you can think of the water column distributions as being affected by physical and biogeochemical processes:
where the space- and time-varying property of interest ( C ) is affected by some physical redistribution processes (the operator "P") and some biogeochemical source/sink/transformation process ( J ). We somehow must deconvolve the physical processes to understand the biogeochemical processes. In this lecture, we will discuss the general character of open ocean transport processes and the concept of turbulent diffusion. The reason why we do this is because the nature of turbulent diffusion is not as clear cut and fundamental as is molecular diffusion. Yet it is an ugly necessity, for it involves a parameterization of processes that we have no hope of resolving directly, and which are so ubiquitous and important, that we cannot ignore. Our hope is to at least make you aware of the philosophical underpinnings of the concept, the strengths and weaknesses of the approach, and the caveats that need be kept in mind when dealing with turbulent diffusivities. At least that way, you may have a better idea of why things appear the way they do.
15.2 The Basic Equation
Our starting point is really rather basic: we start with the Daltonian concept of the conservation of mass. Also, we'll only do this in one dimension. If you are really into pain, you can do this in three. We are assuming that the conservation of mass in one dimension can be written quite simply as
which can be stated in words as "the time rate of change of stuff in a fluid parcel is equal to the spatial rate of change of the (molecular) diffusive flux plus any in situ production". The first term (the one on the left hand side) is often called the lagrangian derivative or the "complete derivative", since it quantifies the time rate of change following the fluid parcel. We have to express our conservation law in this form, since it is only from the perspective of the fluid parcel that we can guarantee the conservation of mass. The second term is simply the molecular diffusive flux divergence (D is the molecular diffusivity, which for most substances is of order 10-9m2/s in water). That is, mass will only accumulate in a fluid parcel if the amount of material diffusing into it from one side exceeds or is less than the amount diffusing out. In order for this to occur, the diffusive flux must change with distance. With the assumption that the molecular diffusivity is constant in space (a reasonable assumption in most systems) the above equation can also be expressed in the possibly more familiar form as
but we'll stick the other form for now. Finally, the last term "J" is there to represent possible source or sink processes, such as biological production/consumption or radioactive production/decay. Its form will remain unspecified here, but may be dependent on space, time, C and possibly other variables.
Now it is more common to view things in a space-fixed frame of reference, rather than a framework which follows fluid parcels. This space-fixed frame is referred to as an eulerian (pronounced "oil-air-ian") frame of reference, and we can translate to this frame of reference by taking partial derivatives
which is simply to state that the rate of change in stuff in a fluid parcel is the sum of the time variation in the overall distribution (the first term) and the downstream change of stuff associated with its displacement. Another way of saying this is to recognize that the change in stuff at a location is the sum of the time rate of change of the distribution summed with the divergence of the advective flux. (The advective flux is just the concentration times the velocity). We can then rewrite our conservation equation as
The above equation is mathematically and physically correct in all respects, and if we can define the terms of this equation with adequate precision in both space and time, we could describe the evolution of C throughout the ocean. The trouble is, we can't.
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