Atmospheric diffusion: when anomalous is normal

By: Omduth Coceal

Our long-term health and quality of life depends on the purity of the air we breathe. It is therefore difficult to overstate the importance of understanding and predicting the dispersion of pollutants in the atmosphere. Both observations and modelling clearly have a key role to play here. But in these days of big data and big supercomputers, it is easy to forget the big ideas upon which our understanding of atmospheric diffusion rests. Theoretical understanding in fact underlies the mathematical models upon which computer simulations are based – and the theory of diffusion is far from complete.

The attempt to understand the process of diffusion has a long history, spanning many fields. In 1855 the German physiologist Adolf Fick, who was interested in the transport of nutrients through biological membranes, published what became known as the “diffusion equation”. This equation allows one to calculate the probability of finding a diffusing particle at a given position at any given time. Fick showed that, on average, such a particle would travel a distance proportional to the square root of the time taken (to be precise, this is the root mean square displacement, but we will just call it the “average” displacement here). This behaviour is so fundamental that it is taken to signify “normal” diffusion. In contrast, a particle moving at constant velocity covers a distance proportional to the time of travel. The reason for the difference is that a particle undergoing diffusion changes its velocity from moment to moment.

Figure 1: A drunken particle undergoing Pearson’s random walk. Pearson wanted to know the probability for the drunkard to end up at a given distance from its starting point. The answer was promptly given by Lord Rayleigh.

Such haphazard motion characterises what is known as a random walk, or drunkard’s walk (see Figure 1). It was introduced as a mathematical problem by the British bio-statistician Karl Pearson in a letter to Nature in 1905 [1] in which he asked:

“A man starts from a point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + dr from his starting point O.”

The solution was given in the very next issue of Nature by Lord Rayleigh [2], who had derived it 25 years earlier while considering the rather different physics problem of adding vibrations of equal frequency and amplitude but random phases. Pearson himself was interested in the spread of malaria by mosquitoes, which he was able to show obeyed the diffusion equation. Hence, the displacement of a mosquito is proportional to the square root of time.

Figure 2: A large number of particles starting from the same point and each undergoing a random walk independently produce a diffusing cloud whose average radius increases as the square root of time.

In the same year, in one of his Annus Mirabilis papers, Albert Einstein investigated the random motion of tiny particles suspended in a liquid first observed by the botanist Robert Brown in 1827. Einstein showed that this so-called Brownian motion could be understood quantitatively by assuming the suspended particles are bombarded by molecules in random thermal motion. His theoretical work enabled Jean Perrin to prove the existence of atoms empirically, which earned Perrin a Nobel prize. Einstein deduced that Brownian motion also obeyed the diffusion equation (see Figure 2).

Establishing a connection between diffusion, the random walk and Brownian motion was only the beginning of the story. Strictly speaking, it applies only to processes that are completely random and uncorrelated from moment to moment, i.e. systems without any memory. This is in sharp contrast to purely deterministic systems governed by Newton’s laws. Many real systems are in fact somewhere in between these two extremes, in the domain of complexity, with a mixture of both random and deterministic components. Turbulent flows fall into this category.

Figure 3: Turbulence consists of a spectrum of eddies of different sizes. These give rise to the phenomenon of anomalous diffusion. In the atmosphere, Richardson showed the process to be superdiffusive.

Atmospheric flows are almost always turbulent. The “molecules” of turbulence are “eddies”, which come in different sizes (see Figure 3). When large eddies of a particular size are dominant they produce coherent motions. The smallest eddies produce random motions. The British fluid-dynamicist G. I. Taylor developed an elegant mathematical theory in 1922 taking into account the role of turbulent eddies in atmospheric diffusion [3]. Unlike the jumps induced by random bombardment of molecules in Brownian motion, a spectrum of eddies produces “diffusion by continuous movements”, the title of Taylor’s 1922 paper. His theory showed that turbulent diffusion only behaves like a Brownian random walk at large times, in the so-called diffusion limit. Physically, that is the regime when a dispersing cloud is large compared to the largest turbulent eddies, which then simply cause random mixing. At short times a different behaviour manifests – the displacement of a particle increases proportionally to the time elapsed. This corresponds to the so-called ballistic regime, when particles are simply carried at the approximately steady velocity produced by the largest eddies. At intermediate times a mixture of the two behaviours would be expected. Unlike “normal” diffusion, atmospheric diffusion has memory. It is anomalous.

The anomalous nature of atmospheric diffusion was first demonstrated experimentally by the British meteorologist L. F. Richardson in 1926 [4]. An experiment was conducted on a windy day in London in which 10,000 balloons were released. Each balloon contained a note asking the finder to call and reveal the time and place where the balloon was found. From this information Richardson was able to deduce that the average separation between two balloons increased as time to the power of 1.5, instead of a half. This is superdiffusion. Anomalous diffusion is normal in the atmosphere.

A century after Einstein, Taylor and Richardson, our understanding of anomalous diffusion is still evolving. Many exciting discoveries have been made, exploiting as well as stimulating new mathematical developments. Examples include seemingly arcane methods such as continuous-time random walks, stochastic differential equations, fractals, and fractional calculus. One thing is clear – there is still as much of a need for fundamental theoretical insights. Many of those insights are in fact relevant to a wide range of otherwise seemingly unrelated applications in physical, life and social sciences. Nature, it seems, enjoys being anomalous.

References:

[1] Pearson, K. (1905), The problem of the random walk. Nature 72, 294. DOI: https://doi.org/10.1038/072294b0.

[2] Rayleigh (1905), The problem of the random walk. Nature 72, 318. DOI: https://doi.org/10.1038/072318a0.

[3] Taylor, G. I. (1922), Diffusion by Continuous Movements. Proceedings of the London Mathematical Society, s2-20, 196-212. DOI:10.1112/plms/s2-20.1.196.

[4] Richardson, L. F. and Walker, G. T. (1926), Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society of London. Series A, 110, 709-739. DOI: https://doi.org/10.1098/rspa.1926.0043.

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