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Eddy diffusion

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.

Eddy diffusion simulation of black fluid parcel in white fluid.[1]

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 (Eddy diffusion § Note on stirring and mixing) 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 edit

The theory of eddy diffusion was originally developed, around the end of the 1910s, by G. I. Taylor[2] and L. F. Richardson[3] 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[4] 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.[5]

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,[6] vertical structure of water in lake bodies,[7] and lowest part of the atmosphere[8] 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[9] to internal gravity waves in the upper atmosphere,[10] from deep sea eddy diffusion and buoyancy[11] to nutrient supply to the surface of the mixed layer in the Antarctic Circumpolar Current.[12]

Mathematical formulation of eddy diffusion edit

Source:[13][14]

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  ,   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  . 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):

 

  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:

 

Likewise for the velocity field:

 

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.  . 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   by definition, one can average the entire equation to eliminate all the turbulent fluctuations  , except in non-linear terms (see Reynolds decomposition, Reynolds stress and Reynolds-averaged Navier–Stokes equations). The non-linear advective term becomes:

 
Upon substitution into the conservation equation:
 

If one pushes the third (turbulent) term of the left hand side to right hand side (into  ), the result is:

 
This equation looks like the equation we started with, apart from (i)   and   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):

 

Define a mean material derivative by:

 

This is the material derivative associated with the mean flow (advective term only contains the laminar part of  ). One can distribute the divergence term on right hand side and use this definition of material derivative:

 
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.   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   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  . Start from Eulerian formulation and use the divergence theorem:

 
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  . 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   (see the section on Gradient based theories). See turbulence modeling for more.

Gradient diffusion theory edit

Example of Eulerian reference system of particles in a box.[15]

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  . 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   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   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  . Hence the model is typically called gradient diffusion (or equivalently down-gradient diffusion).

A rough argument for gradient diffusion edit

Source:[3]

 
Conceptual diagram for a simple derivation of eddy diffusion. An eddy mixes the contents of two fluid regions by injecting streams and filaments back and forth in a quasi-random way. The real process is much more chaotic than a simple spiral suggests.   and   stand for the concentrations of the same arbitrary substance that is being mixed by the eddy. The length-scale of the two regions influenced by the eddy in this picture is set by the length-scale of the eddy, and not vice versa.

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  , 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   apart. They contain volume concentrations   and   of an inert, passive tracer. Without loss of generality, let  . Imagine that a single eddy of length scale   and velocity scale   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  . The boundary is perpendicular to the  -axis. The flux from parcel 1 to parcel 2 is then, at least by order of magnitude:

 

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   and  ) is much larger than the eddy length scale  , one can approximate   over   as the derivative of concentration in a continuously varying medium:

 

Based on similarity with Fick's law of diffusion one can interpret the term in parentheses as a diffusion coefficient   associated with this turbulent eddy, given by a product of its length and velocity scales.

 

using a one-dimensional form of continuity equation  , we can write:

 

If   is assumed to be spatially homogeneous, it can be pulled out of the derivative and one gets a diffusion equation of the form:

 

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   is:

 

By comparison with Gaussian distribution, one can identify the variance as   and standard deviation as  , 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 edit

Source:[13]

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:

 
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:
 
This allows the concentration equation to be rewritten as
 
This is again similar to the initial concentration equation, with transformations   and  . 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   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   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   is a true constant.[16]   needs to be at least spatially uniform for it to be possible to write:

 
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,[17] so a more general form like   is permissible. The terminology in scientific articles is not always consistent in this respect.

Shortcomings and counterexamples of the gradient model edit

Gradient models were historically the first models of eddy diffusion.[13] 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:

  1. For a simple case of homogeneous turbulent shear flow [5] the angle between   and   was found to be 65 degrees. Fickian diffusion predicts 0 degrees.
  2. On the sea, surface drifters initially farther apart have higher probability of increasing their physical distance by large amounts than those initially closer. In contrast Fickian diffusion predicts that the change in mutual distance (i.e. initial distance subtracted from the final distance) of the two drifters is independent of their initial or final distances themselves. This was observed by Stommel in 1949.[17]
  3. Near a point source (e.g. a chimmey), time-evolution of the envelope of diffusing cloud of water vapour is typically observed to be linear in time. Fickian diffusion would predict a square root dependence in time,.[4][7]

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 edit

Example of Lagrangian reference system. The observer follows the particle in its path.[15]

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"[18] and later developed in his paper "Statistical theory of turbulence".[19] 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,   is then conserved following a fluid particle and consequently the evolution of the mean field   can be determined from the statistics of the motion of fluid particles.

Lagrangian formulation edit

Source:[13]

Consider an unbounded turbulent flow in which a source at the time   determines the scalar field to some value:

 
  is the position at time   of the fluid particle originating from position   at time t.

If molecular diffusion is neglected,   is conserved following a fluid particle. Then, the value of   at the initial and final points of the fluid particle trajectory are the same:

 
Calculating the expectation of the last equation yields
 

where   is the forward probability density function of particle position.

Dispersion from a point source edit

For the case of a unit point source fixed at location  , i.e.,  , the expectation value of   is

 
This means that the mean conserved scalar field resulting from a point source is given by the probability density function of the particle position   of the fluid particles that originate at the source.

The simplest case to consider is dispersion from a point source, positioned at the origin ( ), 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:

  •  
    Samples of fluid particle paths given by the Langevin equation for times much shorter than Lagrangian time scale. Note that the expectation evolves linearly. (Both axis are expressed in suitable dimensionless quantities).
     
    Samples of fluid particle paths given by the Langevin equation for times much longer than Lagrangian time scale. Note that the expectation evolves as a square root of time. (Both axis are expressed in suitable dimensionless quantities).
    Given that the isotropic turbulent velocity field has zero mean, fluid particles disperse from the origin isotropically, meaning that mean and covariance of the fluid parcel position are respectively
     
     
    where   is the standard deviation and   the Kronecker delta.
  • The standard deviation of the particle displacement is given in terms of the Lagrangian velocity autocorrelation   following by
     
    where   is the root mean square velocity. This result corresponds with the result originally obtained by Taylor.[18]
  • For all times, the dispersion can be expressed in terms of a diffusivity   as
 
  • The quantity   defines a time-scale characteristic of the turbulence called the Lagrangian integral time scale.
  • For small enough times ( ), so that   can be approximated with  , straight-line fluid motion leads to a linear increase of the standard deviation   which, in term, corresponds to a time-dependent diffusivity  . This sheds light onto one of the above stated experimental counterexamples to gradient diffusion, namely the observation of linear spreading rate for smoke near chimney.
  • For large enough times ( ), the dispersion corresponds to diffusion with a constant diffusivity   so that the standard deviation increases as the square root of time following
     
  • This is the same type of dependence as was derived for a simple case of gradient diffusion. This agreement between the two approaches suggests that for large enough times, the gradient model is working well and instead fails to predict the behavior of particles recently ejected from their source.
Langevin equation edit

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  . With this expression for  , the standard deviation of the particle displacement can be integrated to yield

 
According 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 distribution
 
with   given by the previous equation.

Eddy diffusion in natural sciences edit

Eddy diffusion in the ocean edit

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.[20][21]

Vertical transport: overturning circulation and eddy-upwelling edit

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 [20] (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 Antarctic Circumpolar Current is considered to be the dominant source of upwelling, responsible for roughly 80% of its overall intensity.[22] 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.[9] 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.[10]

Horizontal transport: plastics edit

An example of horizontal transport that has received significant research interest in the 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).[23]

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,[24] plankton, jellyfish,[25][12] heat and salt.[11] 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.[23]

Atmosphere edit

Source:[16]

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  , such as smoke through the atmosphere.

For a stationary medium  , 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,

 
Considering a point source, the boundary conditions are
 
where   such that  , where   is the source strength (total amount of   released).

The solution of this problem is a Gaussian function. In particular, the solution for an instantaneous point source of  , with strength  , of an atmosphere in which   is constant,   and for which we consider a Lagrangian system of reference that moves with the mean wind  :

 
Integration 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 edit

K theory has been applied when studying the dynamics of a scalar quantity   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.  , and an infinite crosswind line source, for which, at  

 
Assuming that  , 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   becomes
 
This equation, together with the following boundary conditions
 
where, 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   yield different solutions.
 
Example of a bent-over plume described using K theory in "Diffusion of stack gasses in very stable atmosphere" by Morton L. Barad.[26]

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.[27] 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 edit

In general, K theory comes with some shortcomings. Calder[28] studied the applicability of the diffusion equation to the atmospheric case and concluded that the standard K theory form cannot be generally valid. Monin[29] 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[26] 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 edit

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.

References edit

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eddy, diffusion, fluid, dynamics, eddy, diffusion, eddy, dispersion, turbulent, diffusion, process, which, fluid, substances, together, eddy, motion, these, eddies, vary, widely, size, from, subtropical, ocean, gyres, down, small, kolmogorov, microscales, occu. 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 Eddy diffusion simulation of black fluid parcel in white fluid 1 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 Eddy diffusion Note on stirring and mixing 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 Contents 1 Historical developments 2 Mathematical formulation of eddy diffusion 3 Gradient diffusion theory 3 1 A rough argument for gradient diffusion 3 2 Interpretation from general equations 3 3 Shortcomings and counterexamples of the gradient model 4 Statistical diffusion theory 4 1 Lagrangian formulation 4 1 1 Dispersion from a point source 4 1 1 1 Langevin equation 5 Eddy diffusion in natural sciences 5 1 Eddy diffusion in the ocean 5 1 1 Vertical transport overturning circulation and eddy upwelling 5 1 2 Horizontal transport plastics 5 2 Atmosphere 5 2 1 Atmospheric Boundary Layer 5 2 2 Shortcomings and advantages 6 Note on stirring and mixing 7 ReferencesHistorical developments editThe theory of eddy diffusion was originally developed around the end of the 1910s by G I Taylor 2 and L F Richardson 3 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 4 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 5 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 6 vertical structure of water in lake bodies 7 and lowest part of the atmosphere 8 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 9 to internal gravity waves in the upper atmosphere 10 from deep sea eddy diffusion and buoyancy 11 to nutrient supply to the surface of the mixed layer in the Antarctic Circumpolar Current 12 Mathematical formulation of eddy diffusion editSource 13 14 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 ϕ x t textstyle phi vec x t nbsp x textstyle vec x nbsp 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 ϕ x t textstyle phi vec x t nbsp 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 ϕ t u ϕ K 0 2 ϕ displaystyle frac partial phi partial t nabla cdot vec u phi K 0 nabla 2 phi nbsp K 0 textstyle K 0 nbsp 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 ϕ x t ϕ x t ϕ x t displaystyle phi vec x t langle phi vec x t rangle phi vec x t nbsp Likewise for the velocity field u x t u x t u x t displaystyle vec u vec x t langle vec u vec x t rangle vec u vec x t nbsp 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 ϕ 0 textstyle langle phi rangle 0 nbsp 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 ϕ 0 textstyle langle phi rangle 0 nbsp by definition one can average the entire equation to eliminate all the turbulent fluctuations ϕ textstyle phi nbsp except in non linear terms see Reynolds decomposition Reynolds stress and Reynolds averaged Navier Stokes equations The non linear advective term becomes u ϕ u u ϕ ϕ u ϕ u ϕ displaystyle begin aligned langle vec u phi rangle amp langle left langle vec u rangle vec u right left langle phi rangle phi right rangle amp langle vec u rangle langle phi rangle langle vec u phi rangle end aligned nbsp Upon substitution into the conservation equation ϕ t u ϕ u ϕ K 0 2 ϕ displaystyle frac partial langle phi rangle partial t nabla cdot left langle vec u rangle langle phi rangle langle vec u phi rangle right K 0 nabla 2 langle phi rangle nbsp If one pushes the third turbulent term of the left hand side to right hand side into 2 textstyle nabla 2 nabla cdot nabla nbsp the result is ϕ t u ϕ K 0 ϕ u ϕ displaystyle frac partial langle phi rangle partial t nabla cdot left langle vec u rangle langle phi rangle right nabla cdot left K 0 nabla langle phi rangle langle vec u phi rangle right nbsp This equation looks like the equation we started with apart from i u textstyle vec u nbsp and ϕ textstyle phi nbsp 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 D ϕ D t ϕ u K 0 2 ϕ displaystyle frac D phi Dt phi nabla cdot vec u K 0 nabla 2 phi nbsp Define a mean material derivative by D D t t u displaystyle frac overline D overline D t frac partial partial t langle vec u rangle cdot nabla nbsp This is the material derivative associated with the mean flow advective term only contains the laminar part of u textstyle vec u nbsp One can distribute the divergence term on right hand side and use this definition of material derivative D ϕ D t ϕ u K 0 ϕ u ϕ displaystyle frac overline D langle phi rangle overline D t langle phi rangle nabla cdot langle vec u rangle nabla cdot left K 0 nabla langle phi rangle langle vec u phi rangle right nbsp 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 K 0 ϕ textstyle K 0 nabla langle phi rangle nbsp 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 u ϕ textstyle langle vec u phi rangle nbsp 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 V textstyle V nbsp Start from Eulerian formulation and use the divergence theorem t V ϕ d V K 0 ϕ n d A ϕ u n d A ϕ u n d A displaystyle frac partial partial t int V langle phi rangle text d V oint K 0 nabla langle phi rangle cdot vec n text d A oint langle phi vec u rangle cdot vec n text d A oint langle phi rangle langle vec u rangle cdot vec n text d A nbsp 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 ϕ u textstyle langle phi vec u rangle nbsp 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 ϕ textstyle langle phi rangle nbsp see the section on Gradient based theories See turbulence modeling for more Gradient diffusion theory edit source source source source Example of Eulerian reference system of particles in a box 15 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 ϕ x t 0 textstyle phi vec x t 0 nbsp 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 ϕ t textstyle frac partial phi partial t nbsp 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 ϕ x textstyle phi vec x nbsp 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 ϕ x textstyle nabla phi vec x nbsp Hence the model is typically called gradient diffusion or equivalently down gradient diffusion A rough argument for gradient diffusion edit Source 3 nbsp Conceptual diagram for a simple derivation of eddy diffusion An eddy mixes the contents of two fluid regions by injecting streams and filaments back and forth in a quasi random way The real process is much more chaotic than a simple spiral suggests F 1 textstyle Phi 1 nbsp and F 2 textstyle Phi 2 nbsp stand for the concentrations of the same arbitrary substance that is being mixed by the eddy The length scale of the two regions influenced by the eddy in this picture is set by the length scale of the eddy and not vice versa 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 K 0 textstyle K 0 nbsp 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 D x textstyle Delta x nbsp apart They contain volume concentrations ϕ 1 textstyle phi 1 nbsp and ϕ 2 textstyle phi 2 nbsp of an inert passive tracer Without loss of generality let ϕ 2 gt ϕ 1 textstyle phi 2 gt phi 1 nbsp Imagine that a single eddy of length scale D x textstyle Delta x nbsp and velocity scale U textstyle U nbsp 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 J textstyle J nbsp The boundary is perpendicular to the x textstyle x nbsp axis The flux from parcel 1 to parcel 2 is then at least by order of magnitude J ϕ 1 U ϕ 2 U U D ϕ U D x D ϕ D x displaystyle begin aligned J amp phi 1 U phi 2 U amp U Delta phi amp U Delta x frac Delta phi Delta x end aligned nbsp 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 ϕ 1 textstyle phi 1 nbsp and ϕ 2 textstyle phi 2 nbsp is much larger than the eddy length scale D x textstyle Delta x nbsp one can approximate D ϕ textstyle Delta phi nbsp over D x textstyle Delta x nbsp as the derivative of concentration in a continuously varying medium J U D x ϕ x displaystyle J U Delta x frac partial phi partial x nbsp Based on similarity with Fick s law of diffusion one can interpret the term in parentheses as a diffusion coefficient K textstyle K nbsp associated with this turbulent eddy given by a product of its length and velocity scales J K ϕ x displaystyle J K frac partial phi partial x nbsp using a one dimensional form of continuity equation ϕ t J x 0 textstyle frac partial phi partial t frac partial J partial x 0 nbsp we can write ϕ t x K ϕ x displaystyle frac partial phi partial t frac partial partial x left K frac partial phi partial x right nbsp If K textstyle K nbsp is assumed to be spatially homogeneous it can be pulled out of the derivative and one gets a diffusion equation of the form ϕ t K 2 ϕ x 2 displaystyle frac partial phi partial t K frac partial 2 phi partial x 2 nbsp 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 x 0 textstyle x 0 nbsp is ϕ x t 1 4 p K t exp x 2 4 K t displaystyle phi x t frac 1 sqrt 4 pi Kt exp left frac x 2 4Kt right nbsp By comparison with Gaussian distribution one can identify the variance as s 2 t 2 K t textstyle sigma 2 t 2Kt nbsp and standard deviation as s t 2 K t t 1 2 textstyle sigma t sqrt 2Kt sim t 1 2 nbsp 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 edit Source 13 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 ϕ t u ϕ K 0 ϕ u ϕ displaystyle frac partial langle phi rangle partial t nabla cdot left langle vec u rangle langle phi rangle right nabla cdot left K 0 nabla langle phi rangle langle vec u phi rangle right nbsp 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 ϕ u K x t ϕ displaystyle langle phi vec u rangle K vec x t nabla langle phi rangle nbsp This allows the concentration equation to be rewritten as ϕ t u ϕ K 0 K ϕ displaystyle frac partial langle phi rangle partial t nabla cdot left langle vec u rangle langle phi rangle right nabla cdot left K 0 K nabla langle phi rangle right nbsp This is again similar to the initial concentration equation with transformations ϕ ϕ u u textstyle phi rightarrow langle phi rangle vec u rightarrow langle vec u rangle nbsp and K 0 K 0 K textstyle K 0 rightarrow K 0 K nbsp 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 K textstyle K nbsp 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 K x t textstyle K vec x t nbsp 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 K textstyle K nbsp is a true constant 16 K textstyle K nbsp needs to be at least spatially uniform for it to be possible to write ϕ t u ϕ K 0 K 2 ϕ displaystyle frac partial langle phi rangle partial t nabla cdot left langle vec u rangle langle phi rangle right K 0 K nabla 2 langle phi rangle nbsp 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 17 so a more general form like K x t textstyle K vec x t nbsp is permissible The terminology in scientific articles is not always consistent in this respect Shortcomings and counterexamples of the gradient model edit Gradient models were historically the first models of eddy diffusion 13 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 For a simple case of homogeneous turbulent shear flow 5 the angle between ϕ textstyle nabla langle phi rangle nbsp and ϕ u textstyle langle phi vec u rangle nbsp was found to be 65 degrees Fickian diffusion predicts 0 degrees On the sea surface drifters initially farther apart have higher probability of increasing their physical distance by large amounts than those initially closer In contrast Fickian diffusion predicts that the change in mutual distance i e initial distance subtracted from the final distance of the two drifters is independent of their initial or final distances themselves This was observed by Stommel in 1949 17 Near a point source e g a chimmey time evolution of the envelope of diffusing cloud of water vapour is typically observed to be linear in time Fickian diffusion would predict a square root dependence in time 4 7 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 edit source source source source Example of Lagrangian reference system The observer follows the particle in its path 15 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 18 and later developed in his paper Statistical theory of turbulence 19 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 ϕ textstyle phi nbsp is then conserved following a fluid particle and consequently the evolution of the mean field ϕ textstyle left langle phi right rangle nbsp can be determined from the statistics of the motion of fluid particles Lagrangian formulation edit Source 13 Consider an unbounded turbulent flow in which a source at the time t 0 textstyle t 0 nbsp determines the scalar field to some value ϕ x t 0 ϕ 0 x displaystyle phi vec x t 0 phi 0 vec x nbsp X t Y textstyle vec X t vec Y nbsp is the position at time t 0 textstyle t 0 nbsp of the fluid particle originating from position Y textstyle vec Y nbsp at time t If molecular diffusion is neglected ϕ textstyle phi nbsp is conserved following a fluid particle Then the value of ϕ textstyle phi nbsp at the initial and final points of the fluid particle trajectory are the same ϕ X t Y t ϕ Y t 0 ϕ 0 Y displaystyle phi vec X t vec Y t phi vec Y t 0 phi 0 vec Y nbsp Calculating the expectation of the last equation yields ϕ x t ϕ 0 Y t x f X x t Y ϕ 0 Y d Y displaystyle left langle phi vec x t right rangle left langle phi 0 vec Y t vec x right rangle int f X vec x t vec Y phi 0 vec Y d vec Y nbsp where f X textstyle f X nbsp is the forward probability density function of particle position Dispersion from a point source edit For the case of a unit point source fixed at location Y 0 textstyle vec Y 0 nbsp i e ϕ 0 x d x Y 0 textstyle phi 0 vec x delta vec x vec Y 0 nbsp the expectation value of ϕ x t textstyle phi vec x t nbsp is ϕ x t f X x t Y 0 displaystyle left langle phi vec x t right rangle f X vec x t Y 0 nbsp This means that the mean conserved scalar field resulting from a point source is given by the probability density function of the particle position f X textstyle f X nbsp of the fluid particles that originate at the source The simplest case to consider is dispersion from a point source positioned at the origin Y 0 0 textstyle Y 0 0 nbsp 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 nbsp Samples of fluid particle paths given by the Langevin equation for times much shorter than Lagrangian time scale Note that the expectation evolves linearly Both axis are expressed in suitable dimensionless quantities nbsp Samples of fluid particle paths given by the Langevin equation for times much longer than Lagrangian time scale Note that the expectation evolves as a square root of time Both axis are expressed in suitable dimensionless quantities Given that the isotropic turbulent velocity field has zero mean fluid particles disperse from the origin isotropically meaning that mean and covariance of the fluid parcel position are respectively X t 0 0 t U s 0 d s 0 displaystyle left langle vec X t 0 right rangle int 0 t left langle vec U s 0 right rangle ds 0 nbsp X i t 0 X j t 0 s X 2 t d i j displaystyle left langle X i t 0 X j t 0 right rangle sigma X 2 t delta ij nbsp where s x t textstyle sigma x t nbsp is the standard deviation and d i j textstyle delta ij nbsp the Kronecker delta The standard deviation of the particle displacement is given in terms of the Lagrangian velocity autocorrelation r s textstyle rho s nbsp following by s X 2 t 2 u 2 0 t t s r s d s displaystyle sigma X 2 t 2u 2 int 0 t t s rho s ds nbsp where u textstyle u nbsp is the root mean square velocity This result corresponds with the result originally obtained by Taylor 18 For all times the dispersion can be expressed in terms of a diffusivity G T t textstyle hat Gamma T t nbsp asG T t 1 2 d d t s X 2 u 2 0 t r s d s displaystyle hat Gamma T t frac 1 2 frac d dt sigma X 2 u 2 int 0 t rho s ds nbsp The quantity T L 0 r s d s displaystyle T L int 0 infty rho s ds nbsp defines a time scale characteristic of the turbulence called the Lagrangian integral time scale For small enough times t T L textstyle t ll T L nbsp so that r s textstyle rho s nbsp can be approximated with r 0 1 textstyle rho 0 1 nbsp straight line fluid motion leads to a linear increase of the standard deviation s X u t textstyle sigma X approx u t nbsp which in term corresponds to a time dependent diffusivity G T t u 2 t textstyle hat Gamma T t approx u 2 t nbsp This sheds light onto one of the above stated experimental counterexamples to gradient diffusion namely the observation of linear spreading rate for smoke near chimney For large enough times t T L textstyle t gg T L nbsp the dispersion corresponds to diffusion with a constant diffusivity G T u 2 T L textstyle Gamma T u 2 T L nbsp so that the standard deviation increases as the square root of time followings X t 2 u 2 T L t displaystyle sigma X t approx sqrt 2u 2 T L t nbsp This is the same type of dependence as was derived for a simple case of gradient diffusion This agreement between the two approaches suggests that for large enough times the gradient model is working well and instead fails to predict the behavior of particles recently ejected from their source Langevin equation edit 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 r s exp s T L displaystyle rho s exp s T L nbsp With this expression for r s displaystyle rho s nbsp the standard deviation of the particle displacement can be integrated to yields X 2 t 2 u 2 T L t T L 1 exp t T L displaystyle sigma X 2 t 2u 2 T L t T L 1 exp t T L nbsp According 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 distribution ϕ x t s X 2 p 3 exp x i x i 2 s X 2 displaystyle left langle phi vec x t right rangle sigma X sqrt 2 pi 3 exp x i x i 2 sigma X 2 nbsp with s X t displaystyle sigma X t nbsp given by the previous equation Eddy diffusion in natural sciences editEddy diffusion in the ocean edit 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 20 21 Vertical transport overturning circulation and eddy upwelling edit 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 20 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 Antarctic Circumpolar Current is considered to be the dominant source of upwelling responsible for roughly 80 of its overall intensity 22 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 9 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 10 Horizontal transport plastics edit An example of horizontal transport that has received significant research interest in the 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 23 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 24 plankton jellyfish 25 12 heat and salt 11 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 23 Atmosphere edit Source 16 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 ϕ textstyle phi nbsp such as smoke through the atmosphere For a stationary medium ϕ textstyle phi nbsp 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 ϕ t x K x ϕ x y K y ϕ y z K z ϕ z displaystyle frac partial phi partial t frac partial partial x left K x frac partial phi partial x right frac partial partial y left K y frac partial phi partial y right frac partial partial z left K z frac partial phi partial z right nbsp Considering a point source the boundary conditions are 1 ϕ 0 as t for lt x lt 2 ϕ 0 as t 0 for x 0 displaystyle begin aligned 1 quad amp phi rightarrow 0 quad text as quad t rightarrow infty quad text for quad infty lt x lt infty 2 quad amp phi rightarrow 0 quad text as quad t rightarrow 0 quad text for quad x neq 0 end aligned nbsp where ϕ textstyle phi rightarrow infty nbsp such that ϕ d x F textstyle int infty infty phi dx Phi nbsp where F textstyle Phi nbsp is the source strength total amount of ϕ displaystyle phi nbsp released The solution of this problem is a Gaussian function In particular the solution for an instantaneous point source of ϕ textstyle phi nbsp with strength F textstyle Phi nbsp of an atmosphere in which u textstyle overline u nbsp is constant v w 0 textstyle v w 0 nbsp and for which we consider a Lagrangian system of reference that moves with the mean wind u textstyle overline u nbsp ϕ F 1 4 p K t 1 2 exp x 2 4 K t displaystyle frac phi Phi frac 1 4 pi Kt 1 2 exp left frac x 2 4Kt right nbsp Integration 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 edit K theory has been applied when studying the dynamics of a scalar quantity ϕ displaystyle phi nbsp 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 ϕ t 0 textstyle partial phi partial t 0 nbsp and an infinite crosswind line source for which at z 0 textstyle z 0 nbsp y K y ϕ y 0 displaystyle frac partial partial y left K y frac partial phi partial y right 0 nbsp Assuming that K x ϕ x x u ϕ x textstyle partial K x partial phi partial x partial x ll overline u partial phi partial x nbsp 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 q textstyle q nbsp becomesu ϕ x z K z ϕ z displaystyle overline u frac partial phi partial x frac partial partial z left K z frac partial phi partial z right nbsp This equation together with the following boundary conditions 1 ϕ 0 as z 2 ϕ 0 as x 0 for all z gt 0 but ϕ as x 0 z 0 such that lim x 0 0 u ϕ d z F 3 K z ϕ z as z 0 for all x gt 0 displaystyle begin aligned 1 quad amp phi rightarrow 0 quad text as quad z rightarrow infty 2 quad amp phi rightarrow 0 quad text as quad x rightarrow 0 quad text for all quad z gt 0 quad text but quad phi rightarrow infty quad text as quad x rightarrow 0 quad z rightarrow 0 quad text such that quad lim x rightarrow 0 int 0 infty overline u phi dz Phi 3 quad amp K z frac partial phi partial z quad text as quad z rightarrow 0 quad text for all quad x gt 0 end aligned nbsp where 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 K z textstyle K z nbsp yield different solutions nbsp Example of a bent over plume described using K theory in Diffusion of stack gasses in very stable atmosphere by Morton L Barad 26 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 27 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 edit In general K theory comes with some shortcomings Calder 28 studied the applicability of the diffusion equation to the atmospheric case and concluded that the standard K theory form cannot be generally valid Monin 29 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 26 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 editThe 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 References edit Transport and Diffusion personalpages to infn it Retrieved 2022 04 01 Taylor Geoffrey Ingram Shaw William Napier 1915 01 01 I Eddy motion in the atmosphere Philosophical Transactions of the Royal Society of London Series A Containing Papers of a Mathematical or Physical Character 215 523 537 1 26 Bibcode 1915RSPTA 215 1T doi 10 1098 rsta 1915 0001 a b Cushman Roisin Benoit Jean Marie Beckers 2011 Introduction to geophysical fluid dynamics physical and numerical aspects 2nd ed Waltham MA Academic Press ISBN 978 0 12 088759 0 OCLC 760173075 a b Cushman Roisin Benoit December 2008 Beyond eddy diffusivity an alternative model for turbulent dispersion Environmental Fluid Mechanics 8 5 6 543 549 doi 10 1007 s10652 008 9082 7 ISSN 1567 7419 S2CID 122415464 a b Tavoularis Stavros Corrsin Stanley March 1981 Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient Part 1 Journal of Fluid Mechanics 104 311 347 doi 10 1017 S0022112081002930 ISSN 0022 1120 S2CID 121757951 Kalinske A A Pien C L 1944 Eddy Diffusion Industrial amp Engineering Chemistry 36 3 220 223 doi 10 1021 ie50411a008 ISSN 0019 7866 a b Min I A Lundblad H L 2002 Measurement and Analysis of Puff Dispersion above the Atmospheric Boundary Layer Using Quantitative Imagery Journal of Applied Meteorology 41 10 1027 1041 Bibcode 2002JApMe 41 1027 doi 10 1175 1520 0450 2002 041 lt 1027 MAAOPD gt 2 0 CO 2 ISSN 0894 8763 JSTOR 26185358 Pasquill Frank Taylor Geoffrey Ingram 1949 07 22 Eddy diffusion of water vapour and heat near the ground Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences 198 1052 116 140 Bibcode 1949RSPSA 198 116P doi 10 1098 rspa 1949 0090 S2CID 98779854 a b Marshall J Speer K Closure of the meridional overturning circulation through Southern Ocean upwelling Nature Geosci 5 171 180 2012 a b Jacob Daniel J Introduction to atmospheric chemistry Princeton University Press 1999 a b Dong Changming McWilliams James C Liu Yu Chen Dake 2014 02 18 Global heat and salt transports by eddy movement Nature Communications 5 1 3294 Bibcode 2014NatCo 5 3294D doi 10 1038 ncomms4294 ISSN 2041 1723 PMID 24534770 a b Berline L Zakardjian B Molcard A Ourmieres Y Guihou K 2013 05 15 Modeling jellyfish Pelagia noctiluca transport and stranding in the Ligurian Sea Marine Pollution Bulletin 70 1 90 99 doi 10 1016 j marpolbul 2013 02 016 ISSN 0025 326X PMID 23490349 a b c d Pope Stephen B 2000 08 10 Turbulent Flows Higher Education from Cambridge University Press doi 10 1017 cbo9780511840531 ISBN 9780521591256 Retrieved 2022 03 24 Cushman Roisin Benoit and Jean Marie Beckers Introduction to geophysical fluid dynamics physical and numerical aspects Academic press 2011 a b Lagrangian Turbulence personalpages to infn it Retrieved 2022 03 29 a b Gifford F A Jr 1968 10 31 An Outline of Theories of Diffusion in the Lower Layers of the Atmosphere Report doi 10 2172 4501607 OSTI 4501607 TID 24190 a b Stommel H 1949 Horizontal diffusion due to oceanic turbulence J Mar Res 8 199 225 a b Taylor G I 1922 Diffusion by Continuous Movements Proceedings of the London Mathematical Society s2 20 1 196 212 doi 10 1112 plms s2 20 1 196 ISSN 0024 6115 Taylor Geoffrey Ingram null null 1935 09 02 Statistical theory of turbulenc Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences 151 873 421 444 Bibcode 1935RSPSA 151 421T doi 10 1098 rspa 1935 0158 S2CID 123146812 a b Wunsch Carl 2004 01 21 Vertical Mixing Energy and the General Circulation of the Oceans Annual Review of Fluid Mechanics 36 1 281 314 Bibcode 2004AnRFM 36 281W doi 10 1146 annurev fluid 36 050802 122121 ISSN 0066 4189 Munk Walter Wunsch Carl 1998 12 01 Abyssal recipes II energetics of tidal and wind mixing Deep Sea Research Part I Oceanographic Research Papers 45 12 1977 2010 Bibcode 1998DSRI 45 1977M doi 10 1016 S0967 0637 98 00070 3 ISSN 0967 0637 Talley Lynne 2013 03 01 Closure of the Global Overturning Circulation Through the Indian Pacific and Southern Oceans Schematics and Transports Oceanography 26 1 80 97 doi 10 5670 oceanog 2013 07 ISSN 1042 8275 a b van Sebille Erik Aliani Stefano Law Kara Lavender Maximenko Nikolai Alsina Jose M Bagaev Andrei Bergmann Melanie Chapron Bertrand Chubarenko Irina Cozar Andres Delandmeter Philippe 2020 02 01 The physical oceanography of the transport of floating marine debris Environmental Research Letters 15 2 023003 Bibcode 2020ERL 15b3003V doi 10 1088 1748 9326 ab6d7d ISSN 1748 9326 S2CID 212423141 Budyansky M V Goryachev V A Kaplunenko D D Lobanov V B Prants S V Sergeev A F Shlyk N V Uleysky M Yu 2015 02 01 Role of mesoscale eddies in transport of Fukushima derived cesium isotopes in the ocean Deep Sea Research Part I Oceanographic Research Papers 96 15 27 arXiv 1410 2359 Bibcode 2015DSRI 96 15B doi 10 1016 j dsr 2014 09 007 ISSN 0967 0637 S2CID 119118270 Johnson Donald R Perry Harriet M Graham William M 2005 Using nowcast model currents to explore transport of non indigenous jellyfish into the Gulf of Mexico Marine Ecology Progress Series 305 139 146 Bibcode 2005MEPS 305 139J doi 10 3354 meps305139 ISSN 0171 8630 JSTOR 24869879 a b Barad Morton L 1951 Carter J H Gosline C A Hewson E W Landsberg H eds Diffusion of Stack Gases in Very Stable Atmospheres On Atmospheric Pollution A Group of Contributions Boston MA American Meteorological Society pp 9 14 doi 10 1007 978 1 940033 03 7 2 ISBN 978 1 940033 03 7 retrieved 2022 04 04 Sheppard P A 1960 Turbulent Transfer in the Lower Atmosphere By C H B PRIESTLEY University of Chicago Press 1959 130 pp 28s Journal of Fluid Mechanics 8 4 636 638 doi 10 1017 S0022112060220859 ISSN 1469 7645 S2CID 122096534 Calder K L 1965 On the equation of atmospheric diffusion Quarterly Journal of the Royal Meteorological Society 91 390 514 517 Bibcode 1965QJRMS 91 514C doi 10 1002 qj 49709139011 ISSN 0035 9009 Monin A S 1959 01 01 Atmospheric Diffusion Soviet Phys Uspekhi 2 1 50 58 doi 10 1070 PU1959v002n01ABEH003108 OSTI 4200848 Retrieved from https en wikipedia org w index php title Eddy diffusion amp oldid 1186143429, wikipedia, wiki, book, books, library,

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