The aim of this paper is to present the main aspects of the life of Lewis Fry Richardson and his most important scientific contributions. Of particular importance are the seminal concepts of turbulent diffusion, turbulent cascade and self-similar processes, which led to a profound transformation of the way weather forecasts, turbulent flows and, more generally, complex systems are viewed.

Authors and Affiliations

1. Dipartimento di Fisica, Università degli Studi di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185, Rome, Italy

   Angelo Vulpiani

Corresponding author

Correspondence toAngelo Vulpiani.

Translation by Daniele A. Gewurz.

Appendix 1: Weather forecasting: from Richardson to today

The key idea by Richardson to forecast the weather was correct, but in order to put it in practice it was necessary to introduce one further ingredient that he could not possibly have known. Even using modern computers, integrating numerically the equations of hydrodynamics is not easily done, one of the reasons being the so-called numerical instability, which forces us to use very small integration steps\(\Delta t\). Indeed, in the numerical treatment of partial differential equations it is necessary to introduce a discretisation of space, by means of a grid whose intervals measure\(\Delta x\), and of time (with integration step\(\Delta t\)). For the numerical algorithm to converge to the solution of the partial differential equation,\(\Delta t\) and\(\Delta x\) cannot be assigned arbitrary values, but have to satisfy constraints that depend on the equation under exam and, in general, on the initial conditions. Just to give an idea of the problem, consider the equation (far simpler than the one used by Richardson)

where\(v\) is a constant and\(-\infty < x < \infty \). A simple, but efficient, approximation of this equation is

where\(u_j^n=u(j \Delta x, n \Delta t)\). In this case, in order to get stability for algorithm (1.2), “Courant condition” (not yet known in Richardson’s times) has to hold:

since the equation is linear,\(C\) does not depend on the initial condition.

In the problem studied by Richardson the negative result (the necessity of using a very small\(\Delta t\)) is not just a mathematical detail, but has a precise physical origin. It is a consequence of the presence in the atmosphere of phenomena, such as sound waves and gravity waves, that have very short characteristic times. Directly using the equations of fluid dynamics and thermodynamics is not a promising approach; so, hoping for significant progress, it is necessary to understand which aspects of the problem have to be taken into account and which ones can be ignored.

We have to remark that fast phenomena are not especially interesting for weather forecasting, but they influence the slow variables, so they have to be somehow accounted for. The way to solve the problem was found by Charney and colleagues in the 1940s–1950s, within the Meteorological Project at the Institute for Advanced Study, in Princeton. The project involved scientists from different fields: mathematicians (such as the great John von Neumann), experts in meteorology, engineering, and computer science. We might say that his project marked the beginning of making Richardson’s dream come true.

In this important scientific effort, a fundamental role was played by the achievement of technological advances (such as the construction of the first modern computer, ENIAC), of numerical methods (algorithms for the integration of partial differential equations), and of modelling (that is, the development of effective equations for meteorology).

Charney and his colleagues noticed that the equations originally proposed by Richardson, even though correct, are not suitable for weather forecasting. The apparently paradoxical reason is that they are too accurate: they also describe high-frequency wave motions that are irrelevant for meteorology. So it is necessary to construct effective equations that get rid of the fast variables. The introduction of a sifting procedure that sets the meteorologically significant part apart from the irrelevant one has a clear practical advantage: the numerical instabilities are less severe, and so we can use a relatively large integration step\(\Delta t\), thus obtaining more efficient numerical computations.

In addition to the computational point of view, it is important that the effective equations for the slow dynamics make it possible to detect the most important factors, which on the contrary remain hidden in the detailed description given by the original equations. The equations now used are called quasi-geostrophic; the simplest case is the barotropic one, where pressure only depends on the horizontal coordinates.

As an example of a construction of effective equations for large-scale behaviour, consider the diffusion equation in a single spatial dimension:

where in the diffusion coefficient\(D(x, x / \epsilon )\) there are two spatial scales: a scale\(O(\epsilon )\) and another\(O(1)\). For instance, we may interpret\(D(x,y)\) as periodic in\(y\) with period\(L\). The system (1.4) describes such physical processes as heat conduction in composite materials. We aim at finding an effective equation that holds for longer times on scales much larger than\(\epsilon \), that is, an equation for the function\(\Theta (x,t)\) obtained by “filtering”\(\theta (x,t)\); in practice, we want a moving average\(\Theta (x,t)=\int _0^L\theta (x+z,t)dz/L \). Homogenisation techniques (the term used in physics ismultiscale) show that:

where

Appendix 2: Diffusion is not always Gaussian

In the simplest case of diffusion, where only molecular diffusion is at work, the probability density\(P(x,t)\) evolves according to the well-known Fick’s equation (for the sake of simplicity, we consider the one-dimensional case on an infinite line):

where\(D\) is the diffusion coefficient. If we consider initial conditions\(P(x,0)\) centred around the origin, the asymptotic solution is a Gaussuan curve:

hence\(\langle x^2(t) \rangle \simeq 2 D t\).

Consider now the problem ofrelative diffusion, which involves two particles with positions\(x_1\) and\(x_2\); in the case where no mutual interaction is present and the probability density of each particle evolves according to Fick’s law, it is easy to see that for the variable\(\ell =x_2-x_1\) the probability density follows Fick’s law (2.2) with the only difference that its diffusion coefficient is doubled:

so\(\langle \ell ^2(t) \rangle \simeq 4 D t\).

Now let us ask a more ambitious question: we have a patch of some substance, transported by a velocity field, and to a first approximation let us assume that the substance is “passive”, that is, it does not influence the velocity field. This is not a mere theoretical problem; it has important real-world applications. Just think of a polluting substance in the sea or in the atmosphere: it is very important to understand how the patch gets larger in time. This is exactly the problem of relative diffusion. Of course, Fick’s law in its simpler form (2.2) does not hold anymore: indeed, the motion of the two particles is influenced by the presence of the velocity field of the fluid (air or water). Richardson, with a deeply insightful understanding of turbulence-related transport phenomena, proposed the following equation:

that is, a Fick’s law in which the diffusion coefficient depends on the distance, in order to keep the turbulent velocity field into account. How do we determine\(D(\ell )\)? From data about diffusion in the atmosphere, he guessed (somewhat daringly) the law\(D(\ell ) \sim \ell ^{4/3}\), from which\(\langle \ell ^2(t) \rangle \sim t^3\).

All of this took place in the 1920s, about two decades before the great Soviet mathematician Andrey Nikolaevich Kolmogorov devised (in part, following Richardson’s ideas about a turbulent cascade) the first modern theory of turbulence: in this context it is easy to prove the correctness of Richardson’s conjecture\(D(\ell ) \sim \ell ^{4/3}\). Of course, the importance of his work does not lie in his “lucky guess”\(D(\ell ) \sim \ell ^{4/3}\), but in his understanding of the physical processes leading to a diffusion equation of the form (7).

It is not hard to find an asymptotic solution to (7):

a function quite different from a Gaussian.

Is all of this consistent with reality? We might argue, as suggested for instance by George Batchelor, in a different way, by considering a Fick’s equation with a diffusion coefficient that is constant when\(\ell \) varies, but depends on time; that is,

with\(D(t) \sim t^2\) (when Kolmogorov’s theory is known, this become obvious); this way, we get, as in Richardson’s approach,\(\langle \ell ^2(t) \rangle \sim t^3\), but with a Gaussian probability density:

We therefore have two different approaches that yield the same result regarding\(\langle \ell ^2(t) \rangle \), but predict different probability densities. Now we know that Richardson’s approach (just like Kolmogorov’s in the 1940s) cannot be completely correct, since the velocity field is intermittent (has a multifractal structure). Only at the beginning of twenty-first century have accurate numerical simulations shown that\(P(\ell , t)\) is substantially in accordance with Richardson’s conclusions: the presence of intermittence only introduces small corrections.

Appendix 3: Fractals, self-similarity and large fluctuations

Systematising the insights by Perrin and Richardson requires new mathematical objects (fractals), which at first sight might look like pathological monstrosities, but are on the contrary extremely common in nature.

Consider a regular curve and let us ask how to measure its length\(L\). We might approximate the curve with a broken line consisting of line segments of length\(\ell \), and then take\(\ell \) smaller and smaller, denoting by\(N(\ell )\) the number of segments. If\(\ell \) is small enough, the length\(L\) is about\(N(\ell ) \ell \), hence\(N(\ell )\) is proportional to\(1/\ell \). In the case of a surface, we can “tile” it with\(N(\ell )\) small squares with side\(\ell \); the area\(A\) can be estimated by\(N(\ell )\ell ^2\), hence\(N(\ell )\) is proportional to\(1/\ell ^2\). Analogously, in order to fill a three-dimensional object, we will need a number of small cubes of side\(\ell \) proportional to\(1/\ell ^3\).

From these remarks, the idea of generalising the notion of dimension arises: an object has a fractal dimension\(D_F\) if the number of “cubes” with side\(\ell \) that are necessary to cover the object behaves like

Clearly, for a regular object the fractal dimension is just the usual dimension. Are there objects for which the\(D_F\) is not an integer? The answer is in the affirmative. An example is given by Koch snowflake, whose construction is outlined in Fig.2: take a line segment of length 1 and divide it into three equal parts. The middle part is taken out and substituted with two segments with the same length. Repeat the procedure on each of the four elements we have now obtained, then on the sixteen elements of the next generation, and so on, an infinite number of times. We get a “very angular” curve having fractal dimension\(\ln 4 /\ln 3 \simeq 1.2618\).

Scheme for the construction of Koch snowflake

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This is indeed a self-similar structure: looking at such a figure under any given resolution we are not able to say the scale we are at. The truly important fact is that this kind of behaviour is not a mere artefact of pathological mathematical models. For instance, if we try to measure the length\(L(\ell )\) of a jagged coastline (as first suggested by Richardson), we find that it gets longer and longer the more the resolution\(\ell \) gets smaller: we have a fractal object with\(D_F\) between\(1\) and\(2\); for Koch snowflake,\(L(\ell ) \sim \ell ^{-(\ln 4/ \ln 3 -1)}\).

This kind of “wrinkledness” is very common: it appears in the attractors of chaotic dissipative dynamic systems and in many natural phenomena; for instance, in turbulence (as described by Richardson in his verse about whirls containing smaller whirls that in turn contain even smaller ones and so on) and in the large-scale structure of galaxies.

Two of the most important results in probability theory are the law of large numbers and the central limit theorem, which hold for independent (or weakly dependent) variables having finite variance. What happens in the case of strongly dependent variables or with infinite variance? As an example, consider a sequence of independent random variables\(\{ x_n\}\),\(n=1,2, \dots , N\), each of which has Cauchy probability density

for these variables (with infinite variance) it is possible to prove the following property: the “average”

for each\(N\) (even an arbitrarily large one) has the same probability distribution as the single variable. We are witnessing here a situation which is very different from that in which the variance is finite, where by the law of large numbers, for\(N \rightarrow \infty \),\((x_1+ \dots +x_N)/N\) converges, besides zero-probability cases, to the mean value.

In the case when the central limit theorem holds, we have that for large values of\(N\) the variable

has a limit distribution (the Gaussian one). If the variance is infinite, in order to have a limit distribution we have to consider the variable

for suitable values of\(\alpha \) and\(C_{\alpha }\), which depend on the distribution of the variables (and especially on the behaviour of the tails); for large values of\(N\) the probability distribution of\(q_N\) converges to a limit function (different from the Gaussian). For instance, in the case when for large values of\(|x|\) the probability density is of the form

then\(\alpha =1\) and the limit distribution is Cauchy; clearly, if the variance is finite we have\(\alpha =1/2\) and the limit distribution is Gaussian.

The energy dissipated in a turbulent fluid as a function of time; note that the enlargement of a part is similar to the whole

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Nowadays, the probability distributions with power-law tails and infinite variance are accepted as non-pathological, and widely used in modelling several physical and financial phenomena. It was not at all so in Richardson’s time; he was one of the first, in his studies about armed conflicts, to notice the non-pathological presence of this kind of probability density.

Even in the cases with a finite variance, when strong correlation between variables at different times is present, it is possible to get “wild fluctuations”; an example given by turbulence is shown in Fig.3. We have there a non-Gaussian probability distribution; looking at the evolution of the variable over time, we note an alternation of long intervals with small fluctuations around its mean value with short, strong, intermittent departures.

Translated from the Italian by Daniele A. Gewurz.

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