Eddy diffusion
Eddy diffusion, eddy dispersion, or turbulent diffusion is a process by which substances are mixed in the atmosphere, the ocean or in any fluid system due to eddy motion. In other words, it is mixing that is caused by eddies that can vary in size from subtropical ocean gyres down to the small Kolmogorov microscales. The concept of turbulence or turbulent flow causes eddy diffusion to occur. 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  (see molecular diffusion). 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. 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 . 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. 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 the molecular diffusivity formula_5, since it is usually significantly smaller than eddy diffusivity, and would steer attention away from the eddy mechanism. Consider two neighbouring fluid parcels with their centers formula_32 apart. They contain volume concentrations formula_33 and formula_34 of an inert, passive tracer. Without loss of generality, let formula_35. Imagine that a single eddy of length scale formula_32 and velocity scale formula_37 is responsible for a continuous stirring of material among the two parcels. The tracer flux exchanged through the lateral boundary of the two  parcels is labelled formula_38. The boundary is perpendicular to the formula_39-axis. The flux from parcel 1 to parcel 2 is then, at least by order of magnitude: formula_40 This argument can be seen as a physically motivated dimensional analysis, since it uses solely the length and velocity scales of an eddy to estimate the tracer flux that it generates. If the entire studied domain (thought to contain a large number of such pairs formula_33 and formula_34) is much larger then the eddy length scale formula_32, one can approximate formula_44 over formula_32 as the derivative of concentration in a continuously  varying medium: formula_46 Based on similarity with Fick's law of diffusion one can interpret the term in parentheses as a diffusion coefficient formula_47 associated with this turbulent eddy, given by a product of its length and velocity scales. formula_48 using a one-dimensional form of continuity equation formula_49, we can write: formula_50 If formula_47 is assumed to be spatially homogeneous, it can be pulled out of the derivative and one gets a diffusion equation of the form: formula_52 This is a prototypical example of parabolic partial differential equation. It is also known as heat equation. Its fundamental solution for  a point source at formula_53 is: formula_54 By comparison with Gaussian distribution, one can identify the variance as formula_55 and standard deviation as formula_56, a very typical time dependence for molecular diffusion or random walk. To conclude this subsection, it described how an eddy can stir two surrounding regions of a fluid and how this behaviour gives rise to mathematics described as "gradient model", meaning that diffusive fluxes are aligned with a negative spatial gradient in concentration. It considered a very simple geometry, in which all variations happen along one axis. The argument used only order-of-magnitude scales of  spatial separation and eddy velocity, therefore it was very rough. The next section offers a more rigorous treatment. Interpretation from general equations. This subsection builds on the section on general mathematical treatment, and observes what happens when a gradient assumption is inserted. Recall the Reynolds-averaged concentration equation: formula_14 We make a similar gradient assumption to that which was motivated in the subsection above with tracer length and velocity scales. However the coefficient value needs not be the same as in the above subsection (which was only specified by order of magnitude). The gradient hypothesis reads: formula_58 This allows the  concentration equation to be rewritten as formula_59 This is again similar to the initial concentration equation, with transofmations formula_60 and formula_61. It represents a generalization to Fick's second law (see Fick's laws of diffusion), in presence of turbulent diffusion and advection by the mean flow. That is the reason why down-gradient eddy diffusion models are often referred to as "Fickian", emphasizing this mathematical similarity. Note that the eddy diffusivity formula_47 can in general be a function of space and time, since its value is given by the pattern of eddies that can evolve in time and vary  from place to place. Different assumptions made about formula_63 can lead to different models, with various trade-offs between observations and theory. Sometimes, the term Fickian diffusion is reserved solely for the case when formula_47 is a true constant. formula_47 needs to be at least spatially uniform for it to be possible to write: formula_66 In this case, the sum of molecular and eddy diffusivity can be considered as a new effective viscosity, acting in qualitatively similar way to molecular diffusivity, but significantly increased in magnitude. In the context of this article, the adjective"Fickian" can also be used as an  equivalent to a gradient model, so a more general form like formula_63 is permissible. The terminology in scientific articles is not always consistent in this respect. Shortcomings and counterexamples of the gradient model. Gradient models were historically the first models of eddy diffusion. They are simple and mathematically convenient, but the underlying assumption on purely down-gradient diffusive flux is not universally valid. Here are a few experimental counter-examples: These observations indicate that there exist mechanisms different from purely down-gradient diffusion, and that the qualitative analogy between molecular and eddy diffusion is not perfect. In the coming section  on statistical models, a different way of looking at eddy diffusion is presented. Statistical diffusion theory. The statistical theory of fluid turbulence comprises a large body of literature and its results are applied in many areas of research, from meteorology to oceanography. Statistical diffusion theory originated with G. I. Taylor's (1921) paper titled "Diffusion by continuous movements" and later developed in his paper "Statistical theory of turbulence". The statistical approach to diffusion is different from gradient based theories as, instead of studying the spacial transport at a fixed point in space, one makes use of the Lagrangian reference system  and follows the particles in their motion through the fluid and tries to determine from these the statistical proprieties in order to represent diffusion. Taylor in particular argued that, at high Reynolds number, the spatial transport due to molecular diffusion can be neglected compared to the convective transport by the mean flow and turbulent motions. Neglecting the molecular diffusion, formula_16 is then conserved following a fluid particle and consequently the evolution of the mean field formula_71 can be determined from the statistics of the motion of fluid particles. Lagrangian formulation. Consider an unbounded turbulent flow in which a source at  the time formula_72 determines the scalar field to some value:formula_73formula_74 is the position at time formula_72 of the fluid particle originating from position formula_76 at time t. If molecular diffusion is neglected, formula_16 is conserved following a fluid particle. Then, the value of formula_16 at the initial and final points of the fluid particle trajectory are the same:formula_79Calculating the expectation of the last equation yields formula_80 where formula_81 is the forward probability density function of particle position. Dispersion from a point source. For the case of a unit point source fixed at location formula_82, i.e., formula_83, the expectation  value of formula_84 isformula_85This means that the mean conserved scalar field resulting from a point source is given by the probability density function of the particle position formula_81 of the fluid particles that originate at the source. The simplest case to consider is dispersion from a point source, positioned at the origin (formula_87), in statistically stationary isotropic turbulence. In particular, consider an experiment where the isotropic turbulent velocity field has zero mean. In this setting, one can derive the following results: formula_96 Langevin equation. The simplest stochastic Lagrangian model is the Langevin equation, which provides a model for the velocity  following the fluid particle. In particular, the Langevin equation for the fluid-particle velocity yields a complete prediction for turbulent dispersion. According to the equation, the Lagrangian velocity autocorrelation function is the exponential formula_106. With this expression for formula_107, the standard deviation of the particle displacement can be integrated to yieldformula_108According to the Langevin equation, each component of the fiuid particle velocity is an Ornstein-Uhlenbeck process. It follows that the fluid particle position (ie., the integral of the Ornstein-Uhlenbeck process) is also a Gaussian process. Thus, the mean scalar field predicted by the Langevin equation is the Gaussian  distributionformula_109with formula_110 given by the previous equation. Eddy diffusion in natural sciences. Eddy diffusion in the ocean. Molecular diffusion is negligible for the purposes of material transport across ocean basins. However, observations indicate that the oceans are under constant mixing. This is enabled by ocean eddies that range from Kolmogorov microscales to gyres spanning entire basins. Eddy activity that enables this mixing continuously dissipates energy, which it lost to smallest scales of motion. This is balanced mainly by tides and wind stress, which act as energy sources that continuously compensate for the dissipated energy. Vertical transport: overturning circulation and eddy -upwelling. Apart from the layers in immediate vicinity of the surface most of the bulk of the ocean is stably stratified. In a few narrow, sporadic regions at high latitudes surface water becomes unstable enough to sink deeply and constitute the deep, southward branch of the overturning circulation (see e.g. AMOC). Eddy diffusion, mainly in the Antarctic Circumpolar Current, then enables the return upward flow of these water masses. Upwelling has also a coastal component owing to the Ekman transport, but Antartic Circumpolar Current is considered to be the dominant source of upwelling, responsible for roughly 80% of its overall  intensity. Hence the efficiency of turbulent mixing in sub-Antarctic regions is the key element which sets the rate of the overturning circulation, and thus the transport of heat and salt across the global ocean. Eddy diffusion also controls the upwelling of atmospheric carbon dissolved in upper ocean thousands of years prior, and thus plays an important role in Earth's climate system. In the context of global warming caused by increased atmospheric carbon dioxide, upwelling of these ancient (hence less carbon-rich) water masses while simultaneously dissolving and downwelling present carbon-rich air, causes a net accumulation of carbon  emissions in the ocean. This in turn moderates the climate change, but causes issues such as ocean acidification. Horizontal transport: plastics. An example of horizontal transport that has received significant research interest in 21st century is the transport of floating plastics. Over large distances, the most efficient transport mechanism is the wind-driven circulation. Convergent Ekman transport in subtropical gyres turns these into regions of increased floating plastic concentration (e.g. Great Pacific garbage patch). In addition to the large-scale (deterministic) circulations, many smaller scale processes blur the overall picture of plastic transport. Sub-grid turbulent diffusion adds a stochastic  nature to the movement. Numerical studies are often done involving large ensemble of floating particles to overcome this inherent stochasticity. In addition, there are also more macroscopic eddies that are resolved in simulations and are better understood. For example, mesoscale eddies play an important role. Mesoscale eddies are slowly rotating vortices with diameters of hundreds of kilometers, characterized by Rossby numbers much smaller than unity. Anticyclonic eddies (counterclockwise in the Northern hemisphere) have an inward surface radial flow component, that causes net accumulation of floating particles in their centre. Mesoscale eddies are not only able to hold debris, but to  also transport it across large distances owing to their westward drift. This has been shown for surface drifters, radioactive isotope markers, plankton, jellyfish, heat and salt. Sub-mesoscale vortices and ocean fronts are also important, but they are typically unresolved in numerical models, and contribute to the above-mentioned stochastic component of the transport. Atmosphere. The problem of diffusion in the atmosphere is often reduced to that of solving the original gradient based diffusion equation under the appropriate boundary conditions. This theory is often called the K theory, where the name comes from the diffusivity coefficient K introduced in the  gradient based theory. If K is considered to be constant, for example, it can be thought of as measuring the flux of a passive scalar quantity formula_16, such as smoke through the atmosphere. For a stationary medium formula_16, in which the diffusion coefficients, which are not necessarily equal, can vary with the three spatial coordinates, the more general gradient based diffusion equation states, formula_113Considering a point source, the boundary conditions areformula_114where formula_115 such that formula_116, where formula_117 is the source strength (total amount of formula_118 released). The solution of this problem is a Gaussian function. In particular, the solution for  an instantaneous point source of formula_16, with strength formula_117, of an atmosphere in which formula_121 is constant, formula_122 and for which we consider a Lagrangian system of reference that moves with the mean wind formula_121:formula_124Integration of this instantaneous-point-source solution with respect to space yields equations for instantaneous volume sources (bomb bursts, for example). Integration of the instantaneous-point source equation with respect to time gives the continuous-point-source solutions. Atmospheric Boundary Layer. K theory has been applied when studying the dynamics of a scalar quantity formula_118 through the atmospheric boundary layer. The assumption of constant eddy  diffusivity can rarely be applied here and for this reason it's not possible to simply apply K theory as previously introduced. Without losing generality, consider a steady state, i.e. formula_126, and an infinite crosswind line source, for which, at formula_127formula_128Assuming that formula_129, i.e., the x-transport by mean flow greatly outweighs the eddy flux in that direction, the gradient based diffusion equation for the flux of a stationary medium formula_130 becomesformula_131This equation, together with the following boundary conditionsformula_132where, in particular, the last condition implies zero flux at the ground. This equation has been the basis for many investigations. Different  assumptions on the form of formula_133 yield different solutions.As an example, K theory is widely used in atmospheric turbulent diffusion (heat conduction from the earth's surface, momentum distribution) because the fundamental differential equation involved can be considerably simplified by eliminating one or more of the space coordinates. Having said that, in planetary-boundary-layer heat conduction, the source is a sinusoidal time function and so the mathematical complexity of some of these solutions is considerable. Shortcomings and advantages. In general, K theory comes with some shortcomings. Calder studied the applicability of the diffusion equation to the atmospheric case  and concluded that the standard K theory form cannot be generally valid. Monin refers to K theory as a semi-empirical theory of diffusion and points out that the basic nature of K theory must be kept in mind as the chain of deductions from the original equation grows longer and more involved. That being said, K theory provides many useful, practical results. One of them is the study by Barad where he a K theory of the complicated problem of diffusion of a bent-over stack plume in very stable atmospheres. Note on stirring and mixing. The verb  stirring has a meaning distinct from "mixing". The former stands for a more large scale phenomenon, such as eddy diffusion, while the latter is sometimes used for more microscopic processes, such as molecular diffusion. They are often used interchangeably, including some scientific literature. "Mixing" is often used for the outcome of both, especially in less formal narration. It can be seen in the animation in the introductory section that eddy-induced stirring breaks down the black area to smaller and more chaotic spatial patterns, but nowhere does any shade of grey appear. Two fluids become more and more intertwined, but  they do not mix due to eddy diffusion. In reality, as their interface becomes larger, molecular diffusion becomes more and more efficient and finishes the homogenization by actually mixing the molecules across the boundaries. This is a truly microscopically irreversible process. But even without molecular diffusion taking care of the last step, one can reasonably claim that spatial concentration is altered due to eddy diffusion. In practice, concentration is defined using a very small but finite control volume in which particles of the relevant species are counted. Averaging over such small control volume yields a useful measure of concentration. This  procedure captures well the action of all eddies smaller than the size of the control volume. This allows to formulate equations describing eddy diffusion and its effect on concentration without the need to explicitly consider molecular diffusion.