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Introduction

Diagram of 1D turbulence

Eddy diffusion, eddy dispersion, multipath, or turbulent diffusion is any diffusion process by which substances are mixed in the atmosphere 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 the small Kolmogorov microscales to subtropical Ocean gyre. The size of eddies decreases as kinetic energy is lost, until it reaches a small enough size for viscosity to control, resulting in kinetic energy dissipating into heat. The concept of turbulence or turbulent flow causes eddy diffusion to occur. The theory of eddy diffusion was 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). In absence of any pressure forces that would favour individual molecules to migrate from a higher concentration area to lower concentration area (i.e. down-gradient), 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.

In turbulent flows, on top of mixing by molecular diffusion, eddies form that stir the fluid. This causes fluid parcels from various initial positions, and thus various associated tracer concentrations, to penetrate into fluid regions with different initial concentrations. This causes the fluid properties to homogenise on scale larger than that of eddies responsible for stiring, in a very efficient way compared to 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.

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 (link to section!) follow the motion of fluid particles, and are thus Lagrangian. In addition, mathematical 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

Adolph Fick, a German physiologist, published a paper in 1855 entitled “Über Diffusion”. In this publication he proposed the following idea:

“It is quite natural to suppose that this law for the diffusion of salt in its solvent must be identical with that according to which the diffusion of heat in a conducting body takes place; upon this law Fourier founded his celebrated theory of heat, and it is the same which Ohm applied, with such extraordinary success, to the diffusion of electricity in a conductor.""

Today the statistical theory of fluid turbulence comprises a large and important body of literature, and its results are applied in many areas from oceanography to cosmology. The study of turbulence by this method actually began, however, with the investigation of turbulent diffusion by Taylor (1921) . The statistical approach to the diffusion problem differs considerably from K theory. Instead of studying the material or momentum flux at a fixed space point, one studies the histories of the motion of individual fluid particles and tries to determine from these the statistical properties necessary to represent diffusion.

In his 1921 paper, Taylor announced his finding in his theorem stating that “[...] the rate at which heat is transferred in the direction of the x axis is determined by the rate of increase of the mean value of the square of the distance, parallel to the axis of x, which is moved through by a particle of fluid in time t."

Mathematical formulation of eddy diffusion

Consider a general conservation equation for a passive conserved scalar field ${\textstyle \phi ({\vec {x}},t)}$, the field being a measure of spatial concentration of this tracer. In line with standard Reynolds decomposition, the concentration field can be divided into its mean and fluctuating components:

$$\phi ({\vec {x}},t)=\langle \phi ({\vec {x}},t)\rangle +\phi '({\vec {x}},t)$$

Likewise for velocity field:

$${\vec {u}}({\vec {x}},t)=\langle {\vec {u}}({\vec {x}},t)\rangle +{\vec {u}}'({\vec {x}},t)$$

Note that the mean field is in general a function of space and time as well. The average is taken such that this makes sense. If applicable can be a spatial average for a fluid parcel larger than the size of eddies, but smaller than the size of the domain of interest. Or an ensemble average. In any case, the mean terms represent a laminar component of the flow, that varies in space and time and remains laminar (by definition).

Consider the conservation equation for ${\textstyle \phi ({\vec {x}},t)}$. This is the generalized fluid Continuity equation with a source term on the right hand side, that corresponds to molecular diffusion:

$${\frac {\partial \phi }{\partial t}}+\nabla \cdot ({\vec {u}}\phi )=K_{0}\nabla ^{2}\phi$$

Or in Lagrangian view:

$${\frac {D\phi }{Dt}}+\phi \nabla \cdot {\vec {u}}=K_{0}\nabla ^{2}\phi$$

${\textstyle K_{0}}$ is the coefficient of Molecular diffusion. One can now proceed with Reynolds decomposition ${\textstyle \phi =\langle \phi \rangle +\phi '}$. Using the fact that ${\textstyle \langle \phi '\rangle =0}$ by definition, one can eliminate all the turbulent fluctuations ${\textstyle \phi '}$, except in non-linear terms (see Reynolds decomposition, Reynolds stress and Reynolds-averaged Navier–Stokes equations. The non-linear advective term becomes:

$${\begin{aligned}\langle {\vec {u}}\phi \rangle &=\langle \left(\langle {\vec {u}}\rangle +{\vec {u}}'\right)\left(\langle \phi \rangle +\phi '\right)\rangle \\&=\langle {\vec {u}}\rangle \langle \phi \rangle \ +\langle {\vec {u}}'\phi '\rangle \end{aligned}}$$

Upon substitution into the conservation equation:

$${\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \ +\langle {\vec {u}}'\phi '\rangle \right)=K_{0}\nabla ^{2}\langle \phi \rangle$$

If one pushes the third (turbulent) term on the right hand side (into ${\textstyle \nabla ^{2}}$), we get:

$${\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=\nabla \cdot \left(K_{0}\nabla \langle \phi \rangle -\langle {\vec {u}}'\phi '\rangle \right)$$

This equation looks like the equation we started with, apart from (i) ${\textstyle {\vec {u}}}$ and ${\textstyle \phi }$ became their laminar components, and (ii) the appearance of a new second term on right hand side. It has analogous function to the Reynolds stress term in the Reynolds-averaged Navier–Stokes equations. equation.

For a Lagrangian formulation, define a mean material derivative by

$${\frac {\overline {D}}{{\overline {D}}t}}={\frac {\partial }{\partial t}}+\langle {\vec {u}}\rangle \cdot \nabla$$

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

$${\frac {{\overline {D}}\langle \phi \rangle }{{\overline {D}}t}}+\langle \phi \rangle \nabla \cdot \langle {\vec {u}}\rangle =\nabla \cdot \left(K_{0}\nabla \langle \phi \rangle -\langle {\vec {u}}'\phi '\rangle \right)$$

This equation looks again like the Lagrangian equation that we started with, with the same caveats as before, and the definition of the mean-flow quantity also for the derivative operator.

The interpretation of eddy diffusivity is as follows. ${\textstyle K_{0}\nabla \langle \phi \rangle }$ is the flux of the passive tracer due to molecular diffusion. Its divergence corresponds to the accumulation (if negative) or depletion (if positive) of the tracer concentration due to this effect. One can interpret the ${\textstyle -\langle {\vec {u}}'\phi '\rangle }$ 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.

One can also examine the concentration budget for a small fluid parcel of volume ${\textstyle V}$. Start from Eulerian formulation and use the divergence theorem:

$${\frac {\partial }{\partial t}}\int _{V}\langle \phi \rangle {\text{d}}V=\oint K_{0}\nabla \langle \phi \rangle \cdot {\vec {n}}{\text{d}}A-\oint \langle \phi '{\vec {u}}'\rangle \cdot {\vec {n}}{\text{d}}A-\oint \langle \phi \rangle \langle {\vec {u}}\rangle \cdot {\vec {n}}{\text{d}}A$$

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 ${\textstyle \langle \phi '{\vec {u}}'\rangle }$. 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 ${\textstyle \langle \phi \rangle }$ (see the section on Gradient based theories). See turbulence modeling for more.

Gradient diffusion theory

Introduction to gradient based theories

The simplest model of turbulent diffusion can be constructed by drawing analogy with 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 ${\textstyle \phi ({\vec {x}},t=0)}$. "Inert passive tracer" means that the tracer does not alter dynamic properties such as density or pressure in any way, it just moves with the flow without modifying it. Let there be a small fluid region with higher concentration of the tracer than its surroundings in every direction. It exchanges fluid and consequently the tracer with its surroundings with turbulent eddies, fluctuating currents going back and forth in a seemingly random way. Due to dynamical isotropy of the situation when the tracer is passive, the eddies flowing to the region from its surroundings are statistically the same as those flowing from the region to its surroundings. The key difference is that those flowing outwards carry much more tracer than those carrying the tracer inwards, since the concentration inside the region is initially higher than outside. This effect would result in itself in an equilibration of ${\textstyle \phi ({\vec {x}})}$ over time. During this process, ${\textstyle {\frac {\partial \phi }{\partial t}}}$ would only depend on the local gradient in concentration profile: ${\textstyle \nabla \phi ({\vec {x}})}$, and the diffusive flux would point precisely in the direction of ${\textstyle -\nabla \phi ({\vec {x}})}$, i.e. downgradient. Therefore, using a gradient term for turbulent tracer concentration flux ${\textstyle \langle \phi '{\vec {u}}'\rangle }$ is a plausible assumption.

Accessible one-dimensional argument for gradient diffusion

In this subsection, means are momentarily not indicated explicitly for simplicity of equations. The subsection aims for a simple and heuristic argument. Also for now neglect the molecular diffusivity ${\textstyle K_{0}}$.

Consider two neighbouring fluid parcels with their centers ${\textstyle \Delta x}$ apart. They contain volume concentrations ${\textstyle \phi _{1}}$ and ${\textstyle \phi _{2}}$ of an inert, passive tracer. Without loss of generality, let ${\textstyle \phi _{2}>\phi _{1}}$. Imagine that a single eddy of length scale ${\textstyle \Delta x}$ and velocity scale ${\textstyle U}$ is responsible for a continuous stirring of material among the two parcels. Define the flux ${\textstyle J}$ as the amount of tracer per unit area per unit time, exchanged through the lateral boundary of the two parcels, here considered as perpendicular to the ${\textstyle x}$-axis. The flux from parcel 1 to parcel 2 is then:

$${\begin{aligned}J&=\phi _{1}U-\phi _{2}U\\&=-U\Delta \phi \\&=-(U\Delta x){\frac {\Delta \phi }{\Delta x}}\end{aligned}}$$

If the studied domain is much larger then the eddy size ${\textstyle \Delta x}$, one can approximate ${\textstyle \Delta \phi }$ per this length scale as the derivative of concentration in a continuously varying medium:

$$J=-(U\Delta x){\frac {\partial \phi }{\partial x}}$$

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

$$J=-K{\frac {\partial \phi }{\partial x}}$$

using a one-dimensional form of continuity equation ${\textstyle {\frac {\partial \phi }{\partial t}}+{\frac {\partial J}{\partial x}}=0}$, we can write:

$${\frac {\partial \phi }{\partial t}}={\frac {\partial }{\partial x}}\left(K{\frac {\partial \phi }{\partial x}}\right)$$

If ${\textstyle K}$ is assumed to be spatially homogeneous, it can be pulled out of the derivative and one gets a diffusion equation of the form:

$${\frac {\partial \phi }{\partial t}}=K{\frac {\partial ^{2}\phi }{\partial x^{2}}}$$

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 ${\textstyle x=0}$ is:

$$\phi (x,t)={\frac {1}{\sqrt {4\pi Kt}}}\exp {\left(-{\frac {x^{2}}{4Kt}}\right)}$$

By comparison with Gaussian distribution, one can identify the variance as ${\textstyle \sigma ^{2}(t)=2Kt}$ and standard deviation as ${\textstyle \sigma (t)={\sqrt {2Kt}}\sim t^{1/2}}$.

Interpretation from general equations

Recall the Reynolds-averaged concentration equation:

$${\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=\nabla \cdot \left(K_{0}\nabla \langle \phi \rangle -\langle {\vec {u}}'\phi '\rangle \right)$$

By a similar argument as in the above heuristic section, it can be postulated that:

$$\langle \phi '{\vec {u}}'\rangle =-K({\vec {x}},t)\nabla \langle \phi \rangle$$

This allows the concentration equation to be rewritten as

$${\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=\nabla \cdot \left((K_{0}+K)\nabla \langle \phi \rangle \right)$$

This is again similar to the initial concentration equation, with ${\textstyle \phi \rightarrow \langle \phi \rangle ,{\vec {u}}\rightarrow \langle {\vec {u}}\rangle }$ and ${\textstyle K_{0}\rightarrow K_{0}+K}$. It represents a generalization to Fick’s second law Fick’s laws of diffusion, in presence of turbulent diffusion and advection by the mean flow. Note that the eddy diffusivity ${\textstyle K}$ can in general be 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 ${\textstyle K({\vec {x}},t)}$ can lead to different models, with various trade-offs between fitting the measurements well in a given physical scenario and being drawn from underlying physics.

Sometimes, the term Fickian diffusion is reserved solely for the case when ${\textstyle K}$ is a constant . ${\textstyle K}$ needs to be at least spatially uniform for it to be possible to write:

$${\frac {\partial \langle \phi \rangle }{\partial t}}+\nabla \cdot \left(\langle {\vec {u}}\rangle \langle \phi \rangle \right)=(K_{0}+K)\nabla ^{2}\langle \phi \rangle$$

In the context of this article, "Fickian" diffusion can also be used as an equivalent to a gradient model , so ${\textstyle K({\vec {x}},t)}$ is permissible, where ${\textstyle {\vec {x}}}$ is a position vector in a fixed coordinate system. The terminology in scientific articles is not always consistent in this respect.

Shortcomings of the gradient model

Gradient models were historically the first models . They are simple and convenient, but the underlying assumption on purely down-gradient diffusive flux is not universally valid.

1. For a simple case of homogeneous turbulent shear flow the angle between ${\textstyle -\nabla \langle \phi \rangle }$ and ${\textstyle \langle \phi '{\vec {u}}'\rangle }$ 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 intially closer. In contrast Fickian diffusion predicts that the change in mutual distance (i.e. initial distance substracted from the final distance) of the two drifters is independent of their initial or final distances themselves. This was observed by Stommel in 1949.
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 ,.

Beyond gradient diffusion: historical case study of distance neighbour graphs

An important early argument for non-Fickian diffusion was given by Henry Stommel and Lewis Fry Richardson for horizontal diffusion at sea . Stommel observed how much surface drifters at sea near Scotland separate in a given time as a function of their initial position. Fickian diffusion would predict a change in mutual distance over time that is independent of their initial distance. The results were inconsistent with this, and Stommel used an earlier model developped by Richardson in 1926 to fit his own data using Richardson’s theory. The following text shows how the independence of the separation change on initial separation can be derived from gradient diffusion in a few steps, and then analyses the Richardson’s alternative model.

One way how to test the gradient model and the resulting Fickian equation of eddy diffusion is to study how the distance between adjacent particles varies in time (neighbour distance function). Consider a one-dimensional eddy diffusion equation with zero mean flow and a constant eddy diffusivity ${\textstyle K}$ (using ${\textstyle K(x)}$ would in the end not change the conclusions):

$${\frac {\partial \phi }{\partial t}}=K{\frac {\partial ^{2}\phi }{\partial x^{2}}}$$

Now let us study the solution to this equation for a point source at ${\textstyle x=x_{s}}$. The solution is indicative of the form of the curve describing the probability density of separation of a particle from its starting point ${\textstyle x=x_{s}}$ at ${\textstyle t=0}$:

$$p(x(t)|x_{s}(0))={\frac {1}{\sqrt {4\pi Kt}}}\exp \left({-{\frac {(x-x_{s})^{2}}{4Kt}}}\right)$$

Now consider two particles. Particle 1 starts at ${\textstyle x(t=0)=0}$ and particle 2 at ${\textstyle x(t=0)=l_{0}}$. Then their individual probability densities to be at ${\textstyle x}$ and ${\textstyle x+l_{1}}$ at some general time ${\textstyle t}$ are ${\textstyle {\frac {1}{\sqrt {4\pi Kt}}}\exp \left({-{\frac {x^{2}}{4Kt}}}\right)}$ and ${\textstyle {\frac {1}{\sqrt {4\pi Kt}}}\exp \left({-{\frac {(x+l_{1}-l_{0})^{2}}{4Kt}}}\right)}$, respectively. Their common probability density to be at those locations simultaneously, and hence the probability density to be ${\textstyle l_{1}}$ apart at ${\textstyle t}$ after starting ${\textstyle l_{0}}$ apart at ${\textstyle t=0}$ is then the product of these equations above, integrated through all offsets ${\textstyle x}$:

$${\begin{aligned}p(l_{1}|l_{0})&={\frac {1}{2\pi Kt}}\exp {\left(-{\frac {(l_{1}-l_{0})^{2}}{4Kt}}\right)}\int _{-\infty }^{+\infty }\exp {\left(-{\frac {x^{2}+x(l_{1}-l_{0})}{Kt}}\right)}dx\\&={\frac {1}{2{\sqrt {\pi Kt}}}}\exp {\left(-{\frac {(l_{1}-l_{0})^{2}}{Kt}}\right)}\end{aligned}}$$

This means that Fickian diffusion results in a probability independent of the initial (${\textstyle l_{0}}$) and final (${\textstyle l_{1}}$) distances themselves. The probability only depends on the differences in their initial and final distances. In the ocean the probability of large values of ${\textstyle (l_{1}-l_{0})^{2}}$ actually increases significantly with ${\textstyle l_{0}}$ and ${\textstyle l_{1}}$, also termed as superdiffusivity. This discrepancy is a fundamental difficulty of the Fickian diffusion model. No absolute spatial dependence ${\textstyle K({\vec {x}})}$ can remedy this in a consistent way .

There is a qualitative physical reason why the eddy diffusion coefficient should not be expected to be the same as the distance between two particles changes. In molecular diffusion, motion of each molecule is independent of that of its neighbours. In turbulent flow, there is a correlation pattern between the velocities of fluid parcels. Neighboring fluid parcels move in a positively correlated way. This correlation of movement quickly decays with distance and it goes to zero when sufficiently far away. The reason is that the distance between neighbouring particles is only affected by smallest scale eddies, the larger scale eddies acting only to transport the pair as a whole. As the separation increases, larger and larger eddies begin contribute to move the two particles with respect to each other. This is an explanation for larger turbulent diffusivity at large distances.

One correction has already been proposed by Richardson in 1926 , originally for atmosphere, but applied by Stommel for the oceans. The diffusion equation is not formulated in a fixed coordinate system, but in terms of separation of particles.

$${\frac {\partial \phi }{\partial t}}={\frac {\partial }{\partial l}}\left(K(l){\frac {\partial \phi }{\partial l}}\right)$$

${\textstyle \phi }$ is now interpreted as the number of particle neighbours per unit separation ${\textstyle l}$. This is not the same as taking ${\textstyle K}$ to be a function of absolute position ${\textstyle K({\vec {x}})}$, such diffusion would still be Fickian in a looser, generalized sense. The diffusion coefficient proposed by Richardson can be in principle homogeneous in absolute space (e.g. position of the midpoint of a pair), but depends on the mutual separation of instantaneous positions of the two particles. This dependence is beyond what is described by the gradient model.

If the time interval during which the motion of particles is studied is close enough, so that ${\textstyle K(l)\approx K(l_{0}),l_{0}}$ being the initial particle separation, than we can approximate this to:

$${\frac {\partial \phi }{\partial t}}=K(l_{0}){\frac {\partial ^{2}\phi }{\partial l^{2}}}$$

When the variance of a collection of particle pairs is measured in an experiment, then the diffusion coefficient ${\textstyle K(l_{0})}$ can be determined as a function of ${\textstyle l_{0}}$. All averages are ensemble averages through all particle pairs. The result ${\textstyle \sigma ^{2}=2Kt}$ derived for Fickian diffusion can still be used, and ${\textstyle \sigma =\langle (l-l_{0})^{2}\rangle }$ can be obtained directly from data.

$$K(l_{0})={\frac {\langle (l-l_{0})^{2}\rangle }{2t}}$$

Fitting this formula to observations gives the ${\textstyle K(l_{0})}$ function. With this knowledge, one can compute how an initial cluster of particles in a given turbulent fluid disperses over time. ${\textstyle K(l_{0})}$ is still an empirical function, this argument does not describe the physics leading to its functional form.

From observations, Richardson and Stommel found ${\textstyle K(l_{0})\sim l_{0}^{1}.4}$ and claim that that is comparable to the dispersion of tracers in the atmosphere. The dependence in those cases ranges between ${\textstyle l_{0}^{1}.3}$ and ${\textstyle l_{0}^{1}.5}$ .

The approach described in this subsection, concentrating already on mutual positions of particles, is a step away from the Eulerian gradient model, and towards Lagrangian models.

Statistical diffusion theory

Introduction to statistical models

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" . 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 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, ${\textstyle \phi }$ is then conserved following a fluid particle and consequently the evolution of the mean field ${\textstyle \left\langle \phi \right\rangle }$ 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 ${\textstyle t_{0}}$ determines the scalar field to some value:

$$\phi ({\vec {x}},t_{0})=\phi _{0}({\vec {x}})$$

${\textstyle {\vec {X}}^{+}(t,{\vec {Y}})}$ is defined as the position at time ${\textstyle t_{0}}$ of the fluid particle originating from position ${\textstyle {\vec {Y}}}$ at time t. It is useful also to consider fluid particle trajectories backward in time.

If molecular diffusion is neglected, then ${\textstyle \phi }$ is conserved following a fluid particle. Then the value of ${\textstyle \phi }$ at the initial and final points of the trajectory that the fluid particle follows are the same:

$$\phi ({\vec {X}}^{+}(t,{\vec {Y}}),t)=\phi _{0}({\vec {Y}},t_{0})=\phi ({\vec {Y_{0}}})$$

Calculating the expectation of the last equation yields

$$\left\langle \phi ({\vec {x}},t)\right\rangle =\left\langle \phi _{0}({\vec {Y}}^{+}(t,{\vec {x}})\right\rangle =\int f_{X}({\vec {x}};t|{\vec {Y}})\phi _{0}({\vec {Y}}){\vec {dY}}$$

where ${\textstyle f_{X}}$ is the forward PDF of particle position.

Dispersion from a point source

For the case of a unit point source fixed at location ${\textstyle {\vec {Y_{0}}}}$, i.e., ${\textstyle \phi _{0}({\vec {x}})=\delta ({\vec {x}}-{\vec {Y_{0}}})}$, the expectation value of ${\textstyle \phi ({\vec {x}},t)}$ is

$$\left\langle \phi ({\vec {x}},t)\right\rangle =f_{X}({\vec {x}};t|Y_{0})$$

This means that the mean conserved scalar field resulting from a point source is given by the PDF of position ${\textstyle f_{X}}$ of the fluid particles that originate at the source.

The simplest case to consider is dispersion from a point source, positioned at the origin (${\textstyle Y_{0}=0}$), in statistically isotropic turbulence. In particular, consider an experiment where the isotropic turbulent velocity field has zero mean and is maintained statistically stationary.

In this setting, one can derive the following results:

• 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
  $$\left\langle {\vec {X}}^{+}(t,0)\right\rangle =\int _{0}^{t}=\left\langle {\vec {U}}^{+}(t',0)\right\rangle dt'=0$$
  $$\left\langle X_{i}^{+}(t,0)X_{j}^{+}(t,0)\right\rangle =\sigma _{X}^{2}(t)\delta _{ij}$$
  where ${\textstyle \sigma _{x}(t)}$ is the standard deviation.
• The standard deviation of the particle displacement is given in terms of the Lagrangian velocity autocorrelation ${\textstyle \rho (s)}$ following by
  $$\sigma _{X}^{2}(t)=2u'^{2}\int _{0}^{t}(t-s)\rho (s)ds$$
  where ${\textstyle u'}$ is the r.m.s. velocity. This result coincides with the result originally obtained by Taylor (1921).
• For all times, the dispersion can be expressed in terms of a diffusivity ${\textstyle {\hat {\Gamma }}_{T}(t)}$ as
  $${\hat {\Gamma }}_{T}(t)={\frac {1}{2}}{\frac {d}{dt}}\sigma _{X}^{2}=u'^{2}\int _{0}^{t}\rho (s)ds$$
• For small enough times (${\textstyle t\ll T_{L}}$), so that ${\textstyle \rho (s)}$ can be approximated with ${\textstyle \rho (0)=1}$, straight-line fluid motion leads to a linear increase of the standard deviation ${\textstyle \sigma _{X}\approx u't}$ which, in term, corresponds to a time-dependent diffusivity ${\textstyle {\hat {\Gamma }}_{T}(t)\approx u'^{2}t}$.
• For large enough times (${\textstyle t\gg T_{L}}$), the dispersion corresponds to diffusion with a constant diffusivity ${\textstyle \Gamma _{T}=u'^{2}T_{L}}$ so that the standard deviation increases as the square root of time following
  $$\sigma _{X}(t)\approx {\sqrt {2u'^{2}T_{L}t}}$$

Eddy diffusion in natural sciences

Atmosphere

The problem of atmospheric diffusion, central in the study of the dynamics of the Atmospheric Boundary Layer, 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. If K is constant the diffusion is called Fickian. The K can be though of as measuring the flux of a passive scalar quantity ${\textstyle q}$, such as smoke.

For a stationary medium q, the gradient based diffusion equation states

$${\frac {\partial q}{\partial t}}=K{\frac {\partial ^{2}q}{\partial x^{2}}}={\frac {\partial }{\partial x}}\left(K_{x}{\frac {\partial q}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(K_{y}{\frac {\partial q}{\partial y}}\right)+{\frac {\partial }{\partial z}}\left(K_{z}{\frac {\partial q}{\partial z}}\right)$$

Considering a point source, the boundary conditions are

$${\begin{aligned}(1)\quad &q\rightarrow 0\quad {\text{as}}\quad t\rightarrow \infty \quad {\text{for}}\quad -\infty <x<\infty \\(2)\quad &q\rightarrow 0\quad {\text{as}}\quad t\rightarrow 0\quad {\text{for}}\quad x\neq 0\end{aligned}}$$

where ${\textstyle q\rightarrow \infty }$ such that ${\textstyle \int _{-\infty }^{\infty }qdx=Q}$, where ${\textstyle Q}$ is the source strength (total amount of q released).

The solution of this problem is a Gaussian function, and in particular the solution for an instantaneous point source of ${\textstyle q}$ with strength ${\textstyle Q}$ of an atmosphere in which ${\textstyle {\overline {u}}}$ is constant, ${\textstyle v=w=0}$ and for which we consider a Lagrangian system of reference that moves with the mean wind ${\textstyle {\overline {u}}}$:

$${\frac {q}{Q}}={\frac {1}{(4\pi Kt)^{1/2}}}\exp \left(-{\frac {x^{2}}{4Kt}}\right)$$

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 - K theory

The assumption of constant eddy diffusivity can rarely be applied to the planetary boundary layer, which is characterised by pronounced shear of the mean wind and large variations in the vertical temperature gradient.

Without losing generality, consider a steady state, i.e. ${\textstyle \partial q/\partial t=0}$, and an infinite crosswind line source, for which, at ${\textstyle z=0}$

$${\frac {\partial }{\partial y}}\left(K_{y}{\frac {\partial q}{\partial y}}\right)=0$$

Assuming that ${\textstyle \partial (K_{x}\partial q/\partial x)/\partial x\ll {\overline {u}}\partial q/\partial x}$, i.e., the x-transport by mean flow greatly outweights the eddy flux in that direction, than Eq. [Ktheory] becomes

$${\overline {u}}{\frac {\partial q}{\partial x}}={\frac {\partial }{\partial z}}\left(K_{z}{\frac {\partial q}{\partial z}}\right)$$

This equation with boundary conditions

$${\begin{aligned}(1)\quad &q\rightarrow 0\quad {\text{as}}\quad z\rightarrow \infty \\(2)\quad &q\rightarrow 0\quad {\text{as}}\quad x\rightarrow 0\quad {\text{for all}}\quad z>0\quad {\text{but}}\quad q\rightarrow \infty \quad {\text{as}}\quad x\rightarrow 0,\quad z\rightarrow 0\quad {\text{such that}}\quad \lim _{x\rightarrow 0}\int _{0}^{\infty }{\overline {u}}qdz=Q\\(3)\quad &K_{z}{\frac {\partial q}{\partial z}}\quad {\text{as}}\quad z\rightarrow 0\quad {\text{for all}}\quad x>0\end{aligned}}$$

In particular, the last condition implies zero flux at the ground, has been the basis for many investigations. Different assumptions on the form of ${\textstyle K_{z}}$ give different solutions.

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, that it lost to smallest scales of motion. This is balanced mainly by tides and wind stress. Those are the energy sources that continuously compensate for the dissipated energy .

Vertical transport: overturning circulation

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 Atlantic Circumpolar Current, then enables the return upward flow of these water masses. Upwelling has also a coastal component owing to the Ekman transport, but Atlantic Circumpolar Current is considered dominant, responsible for roughly 80 % of the overall upwelling intensity. As a result, 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.

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 makes 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 less-stochastic, eddy-related processes. For example, mesoscale eddies play an important role. Mesoscale eddies ae 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 [to do add reference (broken)], 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 .