Eddy diffusion, eddy dispersion, or turbulent diffusion is a process by which substances are mixed in the atmosphere, the ocean or in any fluid system due to eddy motion. In other words, it is mixing that is caused by
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid beh ...
that can vary in size from subtropical
ocean gyre
In oceanography, a gyre () is any large system of circulating ocean currents, particularly those involved with large wind movements. Gyres are caused by the Coriolis effect; planetary vorticity, horizontal friction and vertical friction determine ...
s down to the small
Kolmogorov microscales
In fluid dynamics, Kolmogorov microscales are the smallest scales in the turbulent flow of fluids. At the Kolmogorov scale, viscosity dominates and the turbulence kinetic energy is dissipated into thermal energy. They are defined by
where
* is ...
. The concept of
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
or turbulent flow causes eddy diffusion to occur. The theory of eddy diffusion was first developed by Sir
Geoffrey Ingram Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
.
In
laminar flow
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
s, material properties (salt, heat, humidity, aerosols etc.) are mixed by random motion of individual molecules (see
molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...
). 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
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...
, and its mathematical aspect is captured by the
diffusion equation.
In
turbulent flows, on top of mixing by molecular diffusion,
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid beh ...
stir () the fluid. This causes fluid parcels from various initial positions, and thus various associated concentrations, to penetrate into fluid regions with different initial concentrations. This causes the fluid properties to homogenize on scale larger than that of eddies responsible for stirring, in a very efficient way compared to individual molecular motion. In most macroscopic flows in nature, eddy diffusion is several orders of magnitude stronger than molecular diffusion. This sometimes leads to the latter being neglected when studying turbulent flows.
The problem with turbulent diffusion in the atmosphere and beyond is that there is no single model drawn from fundamental physics that explains all its significant aspects. There are two alternative approaches with non-overlapping areas of utility. According to the gradient transport theory, the diffusion flux at a fixed point in the fluid is proportional to the local concentration gradient. This theory is Eulerian in its nature, i.e. it describes fluid properties in a spatially fixed coordinate system (see
Lagrangian and Eulerian specification of a fluid). In contrast, statistical diffusion theories follow the motion of fluid particles, and are thus Lagrangian. In addition, computational approaches may be classified as continuous-motion or discontinuous-motion theories, depending on whether they assume that particles move continuously or in discrete steps.
Historical developments
The theory of eddy diffusion was originally developed, around the end of the 1910s, by
G. I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
and
L. F. Richardson in England and by
W. Schmidt in Austria as a direct generalization of the classical theory of
molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...
. They proposed the idea that the mass effect of the eddies is entirely similar to that of molecules except for a scale difference. This is described as the "gradient model" in a later section, the name derived from the fact that diffusion fluxes are proportional to the local gradient in concentration, just as for molecular diffusion.
Later research (1930s), mainly by
O. G. Sutton, pointed out some problems of the original approach
and put forward the idea that the difference between the eddy structure of a turbulent fluid and the molecular structure of a fluid at rest is more than one of scale.
During the following decades, a number of studies were carried out to experimentally probe the established theory on eddy diffusion, both for the atmosphere and the ocean/lake bodies, mostly finding agreement with the original theory. In particular, experiments on the diffusion of foreign material in a turbulent water stream, vertical structure of water in lake bodies,
and lowest part of the atmosphere found experimental evidence that eddy diffusion is indeed stronger than molecular diffusion and generally obeys the theory originally developed by
G. I. Taylor
Sir Geoffrey Ingram Taylor OM FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, and a major figure in fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as ...
. Some counter-examples to the original gradient theory are given later in the article.
Active research is now focused on the contributions of eddy diffusion to both atmospheric and oceanic known processes. New models and theories were built on the foundation of the original theory to fully describe these processes. In particular, these studies include eddy diffusion mechanisms to explain processes from aerosols deposition
to internal gravity waves in the upper atmosphere,
from deep sea eddy diffusion and buoyancy
to nutrient supply to the surface of the mixed layer in the
Antarctic Circumpolar Current
The Antarctic Circumpolar Current (ACC) is an ocean current that flows clockwise (as seen from the South Pole) from west to east around Antarctica. An alternative name for the ACC is the West Wind Drift. The ACC is the dominant circulation feat ...
.
Mathematical formulation of eddy diffusion
In this section a mathematical framework based on
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
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
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
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
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. 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
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
and
advection
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...
.
Consider the
conservation equation
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...
for
. This is the generalized fluid continuity equation with a source term on the right hand side. The source corresponds to
molecular diffusion
Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) of ...
(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
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
(
mass diffusivity
Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enco ...
).
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
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
, 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
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
* Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "band ...
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
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
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
In fluid dynamics and turbulence theory, Reynolds decomposition is a mathematical technique used to separate the expectation value of a quantity from its fluctuations.
Decomposition
For example, for a quantity u the decomposition would be
u(x,y,z ...
,
Reynolds stress
In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.
Definition
The veloci ...
and
Reynolds-averaged Navier–Stokes equations
The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged
equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged ...
). 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
In fluid dynamics, the Reynolds stress is the component of the total stress tensor in a fluid obtained from the averaging operation over the Navier–Stokes equations to account for turbulent fluctuations in fluid momentum.
Definition
The veloci ...
term in the
Reynolds-averaged Navier–Stokes equations
The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged
equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-averaged ...
.
This was the Eulerian treatment. One can also study this problem in a Lagrangian point of view (absorbing some terms into the
material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
):
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
In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
:
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
Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real life scenarios, including the flow of blood through the cardiovascular system, the airflow o ...
for more.
Gradient diffusion theory
The simplest model of turbulent diffusion can be constructed by drawing an analogy with the probabilistic effect causing the down-gradient flow as a result of motion of individual molecules (molecular diffusion). Consider an inert, passive tracer dispersed in the fluid with an initial spatial concentration
. 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
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
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
, 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 then 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
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion eq ...
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
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
, 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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
. It is also known as
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
. Its
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not ad ...
for a point source at
is:
By comparison with
Gaussian distribution, one can identify the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
as
and
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
as
, a very typical time dependence for molecular diffusion or
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
.
To conclude this subsection, it described how an eddy can stir two surrounding regions of a fluid and how this behaviour gives rise to mathematics described as "gradient model", meaning that diffusive fluxes are aligned with a negative spatial gradient in concentration. It considered a very simple geometry, in which all variations happen along one axis. The argument used only order-of-magnitude scales of spatial separation and eddy velocity, therefore it was very rough. The next section offers a more rigorous treatment.
Interpretation from general equations
This subsection builds on the section on general mathematical treatment, and observes what happens when a gradient assumption is inserted.
Recall the Reynolds-averaged concentration equation:
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 transofmations
and
. It represents a generalization to Fick's second law (see
Fick's laws of diffusion), in presence of turbulent diffusion and
advection
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is al ...
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.
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,
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
Gradient models were historically the first models of eddy diffusion.
They are simple and mathematically convenient, but the underlying assumption on purely down-gradient diffusive flux is not universally valid. Here are a few experimental counter-examples:
# For a simple case of homogeneous turbulent shear flow
the angle between
and
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.
# 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,.
These observations indicate that there exist mechanisms different from purely down-gradient diffusion, and that the qualitative analogy between molecular and eddy diffusion is not perfect. In the coming section on statistical models, a different way of looking at eddy diffusion is presented.
Statistical diffusion theory
The statistical theory of fluid turbulence comprises a large body of literature and its results are applied in many areas of research, from meteorology to oceanography.
Statistical diffusion theory originated with G. I. Taylor's (1921) paper titled "Diffusion by continuous movements"
and later developed in his paper "Statistical theory of turbulence". The statistical approach to diffusion is different from gradient based theories as, instead of studying the spacial transport at a fixed point in space, one makes use of the
Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
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
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
, 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
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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...
of particle position.
Dispersion from a point source
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
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
turbulence. In particular, consider an experiment where the isotropic turbulent velocity field has zero mean.
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
where
is the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
and
the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
.
* The standard deviation of the particle displacement is given in terms of the ''Lagrangian velocity
autocorrelation
Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ...
'' ''
'' following by
where
is the
root mean square velocity. This result corresponds with the result originally obtained by Taylor.
* 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
=
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