Professor Abullah Ider hsa asked a very interesting set of questions regarding turbulent pair diffusion. Since 1926, this subject has been dominated by Richardson's locality hypothesis which says that the pair diffusion process is governed by local correlations of the same size as the pair separation at any given time. This leads to the Richardson 4/3-pair diffusivity scalingD= where d=|x2-x1| is the pair separation distance, and is the ensemble average. This is equivalent to the famous t^3-regine for the pair separation scaling, =t^3.

This subject is very close to my heart, because due to a great coincidence just last month I have submitted 2 papers to the J. Fluid Mechanics challenging Richardson's assumptions and scaling for the first time in nearly a century.

In the first, theory, paper I show through a novel mathematical formulation (based upon the statistical theory of turbulence) that both local and non-local correlations are in fact effective in the turbulent pair diffusion process, and this leads to different scaling laws to that obtained by Richardson.

In the second paper, I use a Langrangian simulation method to test the new theory -- and all the predictions are indeed observed.

Pre-prints of these papers are available for download from my Researchgate link. (Send me an email otherwise.)

It is important to understand that 'anomalous' diffusion, t^{a}, generally refers to single particle diffusion in which one observes sub-diffusion if 0 1. But is this enough to describe pair diffusion as anomalous diffusion?

In my opinion we need to be careful here. Pair diffusion is quite different to 1-particle diffusion. Even if you could 'fit' an anomalous PDF model to pair diffusion, I am not sure that that is the best way of describing it. Turbulent pair diffusion can be adequately described conventionally as a process in which local and non-local turbulent correlations are effective in increasing the rate of separation of particle pairs.

A critical test for an anomalous diffusion model for pair diffusion would be to establish a mathematical model from first principles (not just a 'fit') that can make predictions that could be tested.

Thank you.

Nadeem Malik

KFUPM

Professor Abullah Ider hsa asked a very interesting set of questions regarding turbulent pair diffusion. Since 1926, this subject has been dominated by Richardson's locality hypothesis which says that the pair diffusion process is governed by local correlations of the same size as the pair separation at any given time. This leads to the Richardson 4/3-pair diffusivity scalingD= where d=|x2-x1| is the pair separation distance, and is the ensemble average. This is equivalent to the famous t^3-regine for the pair separation scaling, =t^3.

This subject is very close to my heart, because due to a great coincidence just last month I have submitted 2 papers to the J. Fluid Mechanics challenging Richardson's assumptions and scaling for the first time in nearly a century.

In the first, theory, paper I show through a novel mathematical formulation (based upon the statistical theory of turbulence) that both local and non-local correlations are in fact effective in the turbulent pair diffusion process, and this leads to different scaling laws to that obtained by Richardson.

In the second paper, I use a Langrangian simulation method to test the new theory -- and all the predictions are indeed observed.

Pre-prints of these papers are available for download from my Researchgate link. (Send me an email otherwise.)

It is important to understand that 'anomalous' diffusion, t^{a}, generally refers to single particle diffusion in which one observes sub-diffusion if 0 1. But is this enough to describe pair diffusion as anomalous diffusion?

In my opinion we need to be careful here. Pair diffusion is quite different to 1-particle diffusion. Even if you could 'fit' an anomalous PDF model to pair diffusion, I am not sure that that is the best way of describing it. Turbulent pair diffusion can be adequately described conventionally as a process in which local and non-local turbulent correlations are effective in increasing the rate of separation of particle pairs.

A critical test for an anomalous diffusion model for pair diffusion would be to establish a mathematical model from first principles (not just a 'fit') that can make predictions that could be tested.

Thank you.

Nadeem Malik

KFUPM

Please have a look at the attached file.

Short (and perhaps not very useful) answers to your questions are:

It's superdiffusion;

self-similar distribution means essentially a power law within the inertial interval;

about correlation function - you need to ask more specifically.

As a longer answer, to begin with, I'd not call it a pair diffusion, but rather, a pair dispersion, because while the equation for the PDF is a superdiffusive equation, it is not the whole truth. If it were a true diffusion (irreversible and dissipative by definition) you would not need to study triplets, because two pairs selected out of a triplet would evolve independently from each other. In reality, two pairs of a triplet will evolve close to each other which means that the distance increments are correlated to a certain degree. That's why you need to consider triplets, quadruplets, material lines, material volumes, etc to improve the understanding.

On the other hand, if you are really only interested in the pair dispersion then, as already mentioned, the PDF is described by a superdiffusive equation _assuming_ that the velocity field is Gaussian (the velocity differences v(r,t)-v(r+a,t) follow a Gaussian distribution). In fully turbulent flows, the velocity is not Gaussian, but the mismatch is not that significant: most often, approximation of Gaussianity works quite well. In that case, I have found it convenient and intuitive to consider the pair dispersion as a biased random walk of a quasi-particle, the coordinate of which is the logarithm of the pair distance, c.f. http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.064501

Thank you so much for your answers.

I think that the mean squared separation for the separating pairs can always be described by a power-like law (t cubed in the Richardson picture). Maybe attributing standard or anomalous to a given diffusion process depends on the nature of the power itself. So, is there any other distributions which are not self-similar?

Jaan, please I mean for example, the computation of the rotation rate Lagrangian auto-correlation structure function as a function of tau for rods in turbulent flows, what is the technical interest behind computing such a quantity? (the decreasing functions that we get generally).

You also mentioned in your comment that the PDF is a superdiffusive equation, could you possibly tell us about the other kinds of the equations that PDFs can satisfy? Thanks a lot for providing me with your paper!

best

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Richardson diffusion is definitely a superdiffusion. It can be derived from Kolmogorov law. Richardson diffusion describes Lagrangian properties of turbulent fluids, i.e. the two point separation of particles. It is present not only in Kolmogorov, but also in compressible turbulence, MHD turbulence etc. The power exponents are different, depending on the slope of the turbulence. It has been recently successfully studied in MHD recently. Note, that the Richardson diffusion scaling is only applicable up to the injection scale of turbulence. On scales larger than the turbulence injection scale, one gets back to diffusive behavior with the standard turbulent diffusion coefficient V_t L_inj

Richardson diffusion is important for many astrophysical processes, including fast magnetic turbulent reconnection, propagation and acceleration of cosmic rays, trasport of heat in plasmas. An remarkable phenomenon related to Richardson diffusion is called spontaneious stochasticity. The essence of the phenomenon is that is that the particles get separated within Richardson diffusion over the distance which does not depend on the initial particle separation. In terms of MHD, Richardson diffusion implies that the standard magnetic flux freezing does not hold in turbulent conducting fluids. Richardson diffusion is, unfortunately, still not well known to astrophysical community and this results in numerous erroneous works.

The power law of the correlation functions of turbulent motions is directly related to the power law index describing Richardson diffusion. Turbulence is a self-similar process over its inertial range. This induces power law behavior on the statistics related to it.

For one generalized view of (potential) distributions which are not self-similar, see "Self-dissimilarity as a high dimensional complexity measure" by Wolpert & Macready. (Other metrics also exist for 'dissimilarity', but I am not an information theorist).

Yes it is anomalous difusion, more exactly superdiffusion

More information you can obtain if you see information about Levy flight diffusion

>I think that the mean squared separation for the separating pairs can always be described by a power-like law

Not necessarily. For instance, if you have a 3D Kolmogorov turbulence with an energy input scale L then at at pair distances \Delta larger than L, the law \Delta^2 \propto t^3 is no longer valid. Also, for \Delta smaller than the Kolmogorov scale l, the streamlines are smooth, and the t^3-law is substituted with an exponential growth \Delta \propto e^{\lambda t}, where \lambda is the largest Lyapunov exponent. While the exponential growth is also a self-similar process (there is no characteristic scale!), the presence of the cross-over scale l breaks the self-similarity.

We say that diffusion is normal if the rms distance grows linearly in time, and otherwise it is anomalous; if it grows faster than linearly (e.g. t^3), we say that we have a superdiffusion. We can have also a subdiffusive beahvior, for instance for a random walk with traps, which is referred to as a continuous time random walk, CTRW. A good example is an array of stationary vortices (with streamfunction \psi = sin x sin y), in which case the dye particles can propagate only along the sepratrix, and interior parts of vortices serve as traps; in that case, \Delta^2 \propto t^{1/2}. If you want to describe such a process with a differential equation for PDF, it will have a fractional time derivative (d/dt)^{1/2} (which is essentially a convolution integral).

As for your autocorrelation functions, there are several factors which will have an impact on it, e.g. large slow vortices will shape the long-time decay of it, but you can also try to reveal the role of persistent structures, such as vortex filaments: if a rod is trapped in a long-living vortex, the rotation rate will remain fairly constant, which contributes to the "fatness" of the long-time tail of the autocorrelation function.

Thanks Jaan a lot for your powerful explanations! I can see things better now!

As everyone has said, it is anomalous.

The term "anomalous" means different things to different people though, when speaking about transport (such as diffusion).

I am not sure of the origin of the term - perhaps others can fill me in on that - but the current convention seems to be that "anomalous diffusion" is synonymous with anything that does not behave like a standard random walk with the same exponent (linear growth of mean square distance versus time). That includes but is not limited to superdiffusion and subdiffusion.

There is another sense of the term however, that is subtly different. That is whether or not the diffusion (and here I am speaking of diffusion in a gas) can be found from classical transport theory, in which one assumes a small deviation from local thermodynamic equilibrium (taking moments of the Maxwell-Boltzmann distribution), and one finds currents of quantities (such as the concentration of a passive dye tracer) that are linearly proportional to the local gradients of quantities (e.g. the dye concentration). Examples include self-diffusion, viscosity, thermal conductivity, etc. Transport that differs from these classical values is "anomalous transport," and that includes diffusion.

In practice, say in plasmas or even just hydrodynamic turbulence, the anomalous transport (if it exists) usually exceeds the classical transport by orders of magnitude. So turbulent "diffusion" mixes dye far, far better than ordinary classical diffusion. It also happens to be superdiffusive, which is why approximating it by an effective diffusion is really really bad, but everybody does it anyway.

I am not a fusion person so others can correct me if I am wrong, but that is my understanding.

Thank you all for your helpful answers! However, I think that the anomalous diffusion should be governed by a Fukker-Planck equation and then we get a stretched exponential PDF. It seems that Richardson used to describe the evolution of the PDF a classical diffusion-like equation (please see here enclosed). How can we say that we have an anomalous diffusion while we use a diffusion-like equation instead of a Fukker-Planck equation? Thanks!

I would like to add an aside if it is not too inappropriate. It is inspired by Prof. Malik's initial reply. A. Lazarian said "Richardson diffusion is definitely a superdiffusion. It can be derived from Kolmogorov law." It is important to distinguish between "Richardson diffusion" and "Richardson's t^3 law." The ~ t^3 relation is indeed a direct consequence of the Kolmogorov 1941 similarity and universality hypotheses applied to relative motion between pairs of particles. However, the t^3 law is consistent with a wide range of models and theories about the processes underlying turbulent dispersion. It can arise from Richardson's d^(4/3) diffusivity (which is why it bears his name; the t^3 law first appeared at the very end of his 1926 paper), or a time-dependent diffusivity (Batchelor Proc. Cambridge Phil. Soc. 1952), or Levy flights or other processes. K41 doesn't say anything about how ~t^3 happens, and we still don't know. Two papers that are related to Prof. Malik's study of whether or not turbulent dispersion is even much of a diffusion process at all include Sawford J. Turbulence 7 (13) (2006) and Klafter, Blumen and Schlesinger Phys. Rev. A 35 (7) (1987). The latter paper also gives some good examples of how non-constant diffusivities can get anomalous diffusion from a "classical diffusion-like equation."

Some confusion can arise from the fact that diffusion is usually defined as a process with the square increment changing in proportion to t. Richardson diffusion is definitely not such sort of a process. It is an accelerated with time process. It has t^3 dependence for Kolmogorov incompressible turbulence or MHD incompressible turbulence, but if the index of turbulence scaling is different, the time dependence is also different.

Hi everybody, although I like the question posed by Abdallah due to its challenging character, it is possibly a bit too general. Also, there is no such thing like a Fukker-Planck equation (well, better than "ck" for "kk"...).

The name is Fokker-Planck equation and represents a generalized heat-conduction equation. I mention heat because heat was the first application of a parabolic equation with constant coefficients used by Fourier to explain the vertical temperature distribution in the Earth (forced by the ~sinusoidal daily temperature dynamics). Later on this parabolic equation has been generalized by guys like Langevin, Fokker, Planck, Einstein and found a microscopic interpretation.

In my view it is somewhat misleading to talk too much about too general concepts, at least in hydrodynamics. There are too many mechanisms in fluids, e.g. remember so-called smart matter, smart fluids like the shear-thinning and shear-thickening fluids, suspensions, solutions etc. It would be more interesting if we would have a plain practical problem, observations of special behaviour. General discussions have been made in length already in the past and classifications are made, all this has been mentioned already in the above contributions. If you, dear Abdallah, want to discover new phenomenae, then you possibly need to become more specific. Cheers, Helmut

Thank you to everyone for a very good discussion. Let me just summaries a few tings.

I agree that the term 'anomalous' diffusion is perhaps a bit too general, essentially referring to anything that is not the standard random walk, or Taylor diffusion case which yields = t. I first heard of the term with reference to Levy flights, which can be modeled through fractional diffusion equations are based upon the idea that waiting times and jump lengths are non-Gaussian.

I would like comment on Jason's statement that, "The ~ t^3 relation is indeed a direct consequence of the Kolmogorov 1941 similarity and universality hypotheses applied to relative motion between pairs of particles."

Essential in this line of argument is the idea of 'locality' which is how Richardson diffusion is most commonly expressed in the turbulence community. Locality means that the pair diffusion process is governed directly by the form of the energy spectrum at a given wavelength k (=1/Del ; Del=pair separation); this is equivalent to saying that the pair diffusion is sensitive to only to local correlations over distances of the order of Del itself. Simple scaling arguments then lead directly to = t^3, This is equivalent to the diffusivity scaling like, D= ^{4/3} (whiuch is how Richardson put it).

However, such this year I have proposed a new theory (under review, JFM) in which locality is not assumed, i.e. pair diffusion is governed by local and non-local correlations. This lead to different power-law scalings for D and . I argue that locality/non-locality is more natural that locality which after all is an ad hoc (and unproven) hypothesis.

Best wishes

Nadee,