Mixing of fluids due to eddy currents
In fluid dynamics, eddy diffusion, eddy dispersion, or turbulent diffusion is a process by which fluid substances mix together due to eddy motion. These eddies can vary widely in size, from subtropical ocean gyres down to the small Kolmogorov microscales, and occur as a result of turbulence (or turbulent flow). The theory of eddy diffusion was first developed by Sir Geoffrey Ingram Taylor.
In laminar flows, material properties (salt, heat, humidity, aerosols etc.) are mixed by random motion of individual molecules. By a purely probabilistic argument, the net flux of molecules from high concentration area to low concentration area is higher than the flux in the opposite direction. This down-gradient flux equilibrates the concentration profile over time. This phenomenon is called molecular diffusion, and its mathematical aspect is captured by the diffusion equation.
In turbulent flows, on top of mixing by molecular diffusion, eddies stir () the fluid. This causes fluid parcels from various initial positions, and thus various associated concentrations, to penetrate into fluid regions with different initial concentrations. This causes the fluid properties to homogenize on scale larger than that of eddies responsible for stirring, in a very efficient way compared to individual molecular motion. In most macroscopic flows in nature, eddy diffusion is several orders of magnitude stronger than molecular diffusion. This sometimes leads to the latter being neglected when studying turbulent flows.
The problem with turbulent diffusion in the atmosphere and beyond is that there is no single model drawn from fundamental physics that explains all its significant aspects. There are two alternative approaches with non-overlapping areas of utility. According to the gradient transport theory, the diffusion flux at a fixed point in the fluid is proportional to the local concentration gradient. This theory is Eulerian in its nature, i.e. it describes fluid properties in a spatially fixed coordinate system (see Lagrangian and Eulerian specification of a fluid). In contrast, statistical diffusion theories follow the motion of fluid particles, and are thus Lagrangian. In addition, computational approaches may be classified as continuous-motion or discontinuous-motion theories, depending on whether they assume that particles move continuously or in discrete steps.
Historical developments.
The theory of eddy diffusion was originally developed, around the end of the 1910s, by G. I. Taylor and L. F. Richardson in England and by W. Schmidt in Austria as a direct generalization of the classical theory of molecular diffusion. They proposed the idea that the mass effect of the eddies is entirely similar to that of molecules except for a scale difference. This is described as the "gradient model" in a later section, the name derived from the fact that diffusion fluxes are proportional to the local gradient in concentration, just as for molecular diffusion.
Later research (1930s), mainly by O. G. Sutton, pointed out some problems of the original approach and put forward the idea that the difference between the eddy structure of a turbulent fluid and the molecular structure of a fluid at rest is more than one of scale.
During the following decades, a number of studies were carried out to experimentally probe the established theory on eddy diffusion, both for the atmosphere and the ocean/lake bodies, mostly finding agreement with the original theory. In particular, experiments on the diffusion of foreign material in a turbulent water stream, vertical structure of water in lake bodies, and lowest part of the atmosphere found experimental evidence that eddy diffusion is indeed stronger than molecular diffusion and generally obeys the theory originally developed by G. I. Taylor. Some counter-examples to the original gradient theory are given later in the article.
Active research is now focused on the contributions of eddy diffusion to both atmospheric and oceanic known processes. New models and theories were built on the foundation of the original theory to fully describe these processes. In particular, these studies include eddy diffusion mechanisms to explain processes from aerosols deposition to internal gravity waves in the upper atmosphere, from deep sea eddy diffusion and buoyancy to nutrient supply to the surface of the mixed layer in the Antarctic Circumpolar Current.
Mathematical formulation of eddy diffusion.
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In this section a mathematical framework based on continuity equation is developed to describe the evolution of concentration profile over time, under action of eddy diffusion. Velocity and concentration field are decomposed into mean and fluctuating (eddy) components. It is then derived that the concentration flux due to eddies is given by covariance of fluctuations in velocity and concentration. This covariance is in principle unknown, which means that the evolution equation for concentration profile cannot be solved without making additional assumptions about the covariance. The next section then provides one such assumption (the gradient model) and thus links to the main result of this section. The one after that describes an entirely different statistical (and Lagrangian) approach to problem.
Consider a scalar field formula_1, formula_2 being a position in a fixed Cartesian coordinate system. The field measures the concentration of a passive conserved tracer species (could be a coloured dye in an experiment, salt in the sea, or water vapour in the air). The adjective "passive" means that, at least within some approximation, the tracer does not alter dynamic properties such as density or pressure in any way. It just moves with the flow without modifying it. This is not strictly true for many "tracers" in nature, such as water vapour or salt. "Conserved" means that there are no absolute sources or sinks, the tracer is only moved around by diffusion and advection.
Consider the conservation equation for formula_1. This is the generalized fluid continuity equation with a source term on the right hand side. The source corresponds to molecular diffusion (and not to any net creation/destruction of the tracer). The equation is written in Eulerian view (it contains partial time derivate):
formula_4
formula_5 is the coefficient of molecular diffusivity (mass diffusivity).
The objective is to find out how the laminar mean flow interacts with turbulent eddies, in particular what effect this has on transport of the tracer. In line with standard Reynolds decomposition, the concentration field can be divided into its mean and fluctuating components:
formula_6
Likewise for the velocity field:
formula_7
The mean term (in angular brackets) represents a laminar component of the flow. Note that the mean field is in general a function of space and time, and not just a constant. Average in this sense does not suggest averaging over all available data in space and time, but merely filtering out the turbulent motion. This means that averaging domain is restricted to an extent that still smoothens the turbulence, but does not erase information about the mean flow itself. This assumes that the scales of eddies and mean flow can be separated, which is not always the case. One can get as close as possible to this by suitably choosing the range of averaging, or ideally doing an ensemble average if the experiment can be repeated. In short, the averaging procedure is not trivial in practice. In this section, the topic is treated theoretically, and it is assumed that such suitable averaging procedure exists. 
The fluctuating (primed) term has the defining property that it averages out, i.e. formula_8. It is used to describe the turbulence (eddies) that, among other things, stirs the fluid.
One can now proceed with Reynolds decomposition. Using the fact that formula_8 by definition, one can average the entire equation to eliminate all the turbulent fluctuations formula_10, except in non-linear terms (see Reynolds decomposition, Reynolds stress and Reynolds-averaged Navier–Stokes equations). The non-linear advective term becomes:
formula_11 Upon substitution into the conservation equation: formula_12
If one pushes the third (turbulent) term of the left hand side to right hand side (into formula_13), the result is: formula_14 This equation looks like the equation we started with, apart from (i) formula_15 and formula_16 became their laminar components, and (ii) the appearance of a new second term on right hand side. This second term has analogous function to the Reynolds stress term in the Reynolds-averaged Navier–Stokes equations.
This was the Eulerian treatment. One can also study this problem in a Lagrangian point of view (absorbing some terms into the material derivative):
formula_17
Define a mean material derivative by:
formula_18
This is the material derivative associated with the mean flow (advective term only contains the laminar part of formula_15). One can distribute the divergence term on right hand side and use this definition of material derivative: formula_20 This equation looks again like the Lagrangian equation that we started with, with the same caveats (i) and (ii) as in Eulerian case, and the definition of the mean-flow quantity also for the derivative operator. The analysis that follows will return to Eulerian picture.
The interpretation of eddy diffusivity is as follows. formula_21 is the flux of the passive tracer due to molecular diffusion. It is always down-gradient. Its divergence corresponds to the accumulation (if negative) or depletion (if positive) of the tracer concentration due to this effect. One can interpret the formula_22 term like a flux due to turbulent eddies stirring the fluid. Likewise, its divergence would give the accumulation/depletion of tracer due to turbulent eddies. It is not yet specified whether this eddy flux should be down-gradient, see later sections.
One can also examine the concentration budget for a small fluid parcel of volume formula_23. Start from Eulerian formulation and use the divergence theorem: formula_24 The three terms on the right hand side represent molecular diffusion, eddy diffusion, and advection with the mean flow, respectively. An issue arises that there is no separate equation for the formula_25. It is not possible to close the system of equations without coming up with a model for this term. The simplest way how it can be achieved is to assume that, just like the molecular diffusion term, it is also proportional to the gradient in concentration formula_26 (see the section on Gradient based theories). See turbulence modeling for more.
Gradient diffusion theory.
The simplest model of turbulent diffusion can be constructed by drawing an analogy with the probabilistic effect causing the down-gradient flow as a result of motion of individual molecules (molecular diffusion). Consider an inert, passive tracer dispersed in the fluid with an initial spatial concentration formula_27. Let there be a small fluid region with higher concentration of the tracer than its surroundings in every direction. It exchanges fluid (and with it the tracer) with its surroundings via turbulent eddies, which are fluctuating currents going back and forth in a seemingly random way. The eddies flowing to the region from its surroundings are statistically the same as those flowing from the region to its surroundings. This is because the tracer is "passive", so a fluid parcel with higher concentration has similar dynamical behaviour as a fluid parcel with lower concentration. The key difference is that those flowing outwards carry much more tracer than those flowing inwards, since the concentration inside the region is initially higher than outside. This can be quantified with a tracer flux. Flux has units of tracer amount per area per time, which is the same as tracer concentration times velocity. Local tracer accumulation rate formula_28 would then depend on the difference of outgoing and incoming fluxes. In our example, outgoing fluxes are larger than ingoing fluxes, producing a negative local accumulation (i.e. depletion) of the tracer. This effect would in general result in an equilibration of the initial profile formula_29 over time, regardless of what the initial profile might be. To be able to calculate this time evolution, one needs to know how to calculate the flux. This section explores the simplest hypothesis: flux is linearly related to the concentration difference (just as for molecular diffusion). This also comes as the most intuitive guess from the analysis just made. Flux is in principle a vector. This vector points in the direction of tracer transport, and in this case it would be parallel to formula_30. Hence the model is typically called gradient diffusion (or equivalently down-gradient diffusion).
A rough argument for gradient diffusion.
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The subsection aims for a simple, rough and heuristic argument explaining how the mathematics of gradient diffusion arises. A more rigorous and general treatment of gradient model is offered in the next subsection, which builds directly on the section on general mathematical treatment (which was not yet assuming gradient model at that early stage and left the covariance of fluctuations as it was). Means are for now not indicated explicitly for maximal simplicity of notation. Also for now neglect