How many angels can dance on the point of a pin?
When I was young this was often quoted as an example of the foolishness of the medieval schoolmen and the nonsensical nature of their discussions. I happily classed those who debated it along with those who, not only believed that the sun was pulled round the heavens in a fiery chariot, but who were quite prepared to specify the precise number of horses pulling the chariot. Later on it seemed that it might have been a sort of reductio ad absurdam, used for critical purposes. Perhaps like the original intention behind Schrodinger’s cat? Later still it seemed that it might be an ironical comment by a seventeenth century protestant theologian. In any case, it has passed into the language as the epitome of foolish and pointless discussion that has some degree of intellectual pretension.

Where then may such pointless intellectual activity be found nowadays? Well, passing over easy targets like the arts, sociology and modern literary criticism, the answer, which may surprise you, is physics. Why should it surprise you? The further answer to that is that physics has been the gift that keeps giving. Over the past century or more, it has given us the impression that it can answer any question, and in the process give rise to amazing developments in science and engineering which alter all our lives for the better. In fact the twentieth-century advances in physics underpin all advances in medicine, transport, engineering and all-round super electronic devices which smooth our paths in so many ways!

As we become less bedazzled by the wonders of quantum theory and relativity, we are more conscious of the inconsistencies, such as dark matter and dark energy, the mysterious use of string theory in many dimensions, and a standard model of the universe which is, in some ways, apparently at a similar stage to the nineteenth-century study of the periodic table, prior to the development of quantum theory. Lee Smolin, in his book The trouble with physics points out the need for a revolution in physics. Roger Penrose in his more recent book Fashion, Faith and Fantasy in the New Physics of the Universe deplores the view that quantum theory has been so successful that it must apply to gravity too. As someone who has always worked in the classical physics area of turbulence theory (albeit using the methods of quantum field theory), I am merely an onlooker. But I have been surprised to notice that much modern physics seems to involve material that I lectured on in statistical field theory to final-year undergraduates and first-year postgraduates. I’m thinking here of topics like mean-field theory and $\phi^4$ scalar field theory. I also tend to feel surprised to see many attempts at a theory of quantum gravity based on the path-integral formulation of quantum mechanics. This is equivalent to solving the Schrödinger equation and one would not do that for a macroscopic box of gas, let alone the universe. Instead, because of the instability of the wave-function, one would use the density matrix formulation.

Every year we turn out thousands of our cleverest young people in all parts of the world to work on cosmology and particle theory. Inevitably their lives are devoted to what can be little more than pedagogical work. In contrast, the important fundamental problems of fluid turbulence receive little attention. I’m not advocating a dirigiste approach of any kind. I very much understand the importance of scholarship and research on fundamentals being a sort of creative ferment. But if a fraction of the effort on lattice QCD went into turbulence simulation, with the same sort of attitudes, it could transform the situation. As it is, we are lumbered with a turbulence community who mostly (it would seem) do not understand the concept of scale-invariance; and therefore do not understand that its onset is what defines the infinite Reynolds number limit!

Academic fathers and Mother Christmas
In the mid-1980s I visited the Max Planck institute in Bonn to give a talk. While I was there, some of the German mathematicians told me about the concept of an academic father. They said that your PhD supervisor was your academic father, his supervisor was your academic grandfather, and so on. In that way: ‘We can all trace our lineage back to Gauss!’

In my own case, Sam Edwards was my supervisor and I was under the impression that Nicholas Kemmer had been his supervisor. Kemmer was retired by the time I joined the Physics department at Edinburgh and I never met him as such. Our only acquaintance was that on his rare visits to the department, he would call hello in passing, as my office door was always open.

I once discussed this concept with colleagues on some social occasion and one of them reckoned that Kemmer’s supervisor had been Weyl. So it turned out that someone I was collaborating with at the time was a sort of academic cousin. I’m not sure just what kind of cousin. My wife is an expert on matters like ‘second cousin, twice removed’, but it’s all Greek to me. Although I’m actually a bit better at Greek that at cousinage.

Recently I checked up on this and found to my surprise that Sam’s supervisor was Julian Schwinger and in turn his had been Isidor Isaac Rabi. This was encouraging, as both were Nobel Laureates in physics. Then Rabi’s supervisor had been Albert Potter Wills, who in turn was supervised by Arthur Gordon Webster (No, me neither.). He at least was supervised by Helmholtz, but after that the trail went cold again and it didn’t look like we were heading back to Gauss.

There must have been some reason why I had thought that Kemmer was Sam’s supervisor. Perhaps that was when he had still been at Cambridge University? Then he would have changed to Schwinger at Harvard? If Kemmer had been Sam’s supervisor for part of the time then he could still count as an ‘academic father’.

So I thought that I would check Kemmer out and found that his supervisor had been Pauli (not Weyl!) and in turn Pauli’s had been Sommerfeld, whose had been Lindemann (the mathematician, not the later physicist), and his had been Klein. Then Klein’s supervisor was Plucker, who was supervised by Gerling and (at last) we are back to Gauss, who was Gerling’s supervisor. But can I claim to be descended from Gauss? Well, I’m still not sure.

Of course this is all a rather old-fashioned idea. There are growing numbers of women in physics and mathematics and if we want to talk about academic descent then we should include academic mothers and, in time, academic grandmothers; and so on. Inclusiveness is the watchword nowadays and as this is Christmas Eve I shall be hanging up my stocking in the hope that Mother Christmas will put some nice presents in it. Certainly she has made a great job of decorating our tree: see below.

If you have been, then thank you for reading; and I wish you a happy Christmas!

Peer Review: Through the Looking Glass
Five years ago, when carrying out direct numerical simulations (DNS) of isotropic turbulence at Edinburgh, we made a surprising discovery. We found that turbulence states died away at very low values of the Reynolds number and the flow became self-organised, taking the form of a Beltrami flow, which has velocity and vorticity vectors aligned. This work is reported in [1] below, and illustrated by the following figure.

A video of the simulation, showing the symmetry-breaking transition, complete with characteristic ‘critical slowing down’, can be found at the online article [1]: https://doi.org/10.1088/1751-8113/48/25/25FT01
The link to the video can be found under the heading Supplementary Data. Downloading this should be straightforward using Windows, but if using a Mac you may have to have an app such as VLC installed.

The article [1] was featured on the front cover of the journal, thus:

It was downloaded hundreds of times within a few days of publication and the total number of downloads now stands at 2708.
That sounds like a success story and you may well wonder why I want to feature this as yet another problem with peer review. The answer to that lies in the fact that we first submitted it to Physical Review Letters and that was such an bizarre experience that it deserves to be told!

Normally I would refer to the two referees as Referee A and Referee B, but as their behaviour seemed to belong to the Looking Glass world (that turbulence assessment so often is) I have decided to call them Tweedledum and Tweedledee.

First, Tweedledum said that he didn’t understand how we were forcing the turbulence. He had never seen anything like that before. Perhaps the strange behaviour was due to our strange forcing. He didn’t think that our Letter should be published.

Then, Tweedledee said that he didn’t understand how we were forcing the turbulence. He also had never seen anything like that before. Perhaps the strange behaviour was due to our strange forcing. He also didn’t think that our Letter should be published.

In Alice Through the Looking Glass, the twins had a famous battle. That did not happen in the present case where they were in perfect agreement; although Tweedledum (or was it Tweedledee?) suggested that perhaps if we did a lot more work and wrote it up as a much longer article, then it might be suitable for publication. This rather misses the point of having a journal like PRL!

When we submitted our paper to J. Phys. A, we pointed out the following: our method of forcing is known as negative damping; it was introduced to turbulence theory in 1965 by Jack Herring; it was first used in DNS in 1997 by Luc Machiels; has subsequently been used in numerous investigations; and in 2005 was studied theoretically by Doering and Petrov [2]. Not precisely an obscure technique then. But what an intellectually feeble performance from Tweedledum and Tweedledee. No wonder problems in turbulence remain unresolved for generations.

One might end up by wondering what if any harm had been done by the lack of scholarly behaviour on the part of these referees who were presumably chosen to be representative of the turbulence community. After all, the paper has been published and has clearly aroused quite a lot of interest. The trouble is, I suspect that J. Phys. A does not have the same visibility among turbulence researchers as PRL. In that case the numerous downloads may reflect the fact that many physicists are interested in an example of a nonlinear phase transition without necessarily having any interest in turbulence. More generally, over the years it seems to me that turbulence referees tend to exert a frictional drag on the process of publishing papers. Many of them give the impression of not wanting the pure pool of ignorance to be spoiled by any new understanding or knowledge.

[1] W. D. McComb, M. F. Linkmann, A. Berera, S. R. Yoffe, and B. Jankauskas. Self-organization and transition to turbulence in isotropic fluid motion driven by negative damping at low wavenumbers. J. Phys. A Math.Theor., 48:25FT01, 2015.
[2] Charles R. Doering and Nikola P. Petrov. Low-wavenumber forcing and turbulent energy dissipation. Progress in Turbulence, 101(1):11-18, 2005.

Should theories of turbulence be intelligible to fluid dynamicists?

One half of the Nobel Prize in physics for 2020 was awarded to Roger Penrose for demonstrating that ‘black hole formation is a robust prediction of the General Theory of Relativity’. While it’s not my field, I do know a little about general relativity; so I had a look at what I could find online. It rapidly became clear to me that in order to understand Penroses’s work in detail, I would have to master a great deal of mathematics – topology in particular – which is unfamiliar to me. This would mean giving up everything else for a substantial period of time and that just wouldn’t make sense. So, despite knowing the basic equations of general relativity (for a simple, yet reasonably complete introduction, see reference [1]), I just have to take the word of other people that it all makes sense.

So what about relativistic quantum field theories, derived from the Navier-Stokes equations? Well, starting with Kraichnan, Wyld and Edwards in the early 1960s and leading up to my own LET theory [2], there exists a moderately successful class of statistical theories of turbulence which are essentially based on quantum field theory. Unfortunately, I would assume that many (most?) fluid dynamicists are as unfamiliar with the background to these as I am with the methods of Penrose in demonstrating that general relativity implies the existence of black holes. Although at least I hope that I belong to the same ‘culture’ as Penrose, in the sense that I appreciate the significance of what he has done and also why he has done it.

The question of how understandable (to turbulence researchers) statistical theories should be, was raised in lecture notes entitled ‘Problems and progress in the theory of turbulence’ [3] by Philip Saffman. In these he wrote down his list of the properties a theory should have. These were generally unexceptionable and really quite obvious. Indeed, one should perhaps bear in mind that a physicist would be very unlikely to write down a similar list, essentially because they would regard it all as being understood. The point that particularly interests me is that the second item in his list, after ‘Clear physical or engineering purpose’ is ‘Intelligibility’. It is worth quoting exactly what he says about this.

‘Intelligibility means that it can be understood, appreciated and applied by a competent scientist without years of study or familiarity with the jargon and techniques of a narrow speciality.’

Obviously, in view of what I wrote at the beginning of this post, I have a certain amount of sympathy with this view. At the same time, I feel that I should challenge it. The final phrase, which I have emphasised, has a faint flavour of the pejorative about it, particularly when taken in conjunction with his other writings. But we are entitled to ask what he means, by a ‘narrow speciality’.

His concern was with those theories of turbulence which are applications of quantum field theory, a subject that made great advances in the 1940s/50s. But quantum field theory was not a ‘narrow speciality’ in the 1970s; and is even less so today. It is a major discipline worldwide and, if we add in statistical field theory in condensed matter physics, then the activity involved would dwarf all turbulence research by orders of magnitude. Moreover, the theory in these areas is closely linked to the experimental work. There is a vast, and growing, body of work in these areas, so this cannot be seen as a narrow or esoteric activity.

Presumably then, he meant simply the applications to turbulence. For Saffman this boiled down to the work of Kraichnan, so he does not give a balanced or scholarly view of this field. Indeed, he does not cite any of the relevant papers by Kraichnan but instead relies on the book by Leslie. It is difficult to see his comments generally as being anything but an expression of frustration that there is an activity going on which he does not understand, combined with a degree of resentment because he felt that his own type of work was somehow being belittled or patronised.

here are other parts of his lecture notes that I value, such as his criticism of Kolmogorov’s 1962 ‘refined theory’; and the general tone of the lectures is undoubtedly stimulating. But although Philip Saffman is no longer here to speak for himself, I still think that his views about fundamental approaches to turbulence should be challenged, if only because similar views seem to be quite widespread today. I am occasionally surprised by how glibly members of the turbulence community are prepared to write off renormalization methods, with phrases such as ‘everyone knows that Kraichnan’s theory is wrong and no one bothers about it anymore’. Well life is so much easier if you pass up on the challenges. But to such people, I would address the question: what have you got to put in its place?

In the mid-1970s, when Saffman was writing, the situation was very different from that today. The basic idea of the LET theory was put forward by me in 1974, incidentally offering a fundamental reason for the failure of the Edwards theory and other cognate theories, including Kraichnan’s. Since then the LET theory has been developed and extensively computed and compared to other theories. I have also published three books, all intended to make such theories more accessible to non-physicists. Two are on turbulence and one on renormalization methods; and their titles can be found in the list of my publications in this blog. So I would like to answer my own question by saying that turbulence theories are intelligible to fluid dynamicists, provided that they are open minded and are prepared to make a bit of an effort. That’s what I would like to say but I have to make one caveat. There are theories, supposedly of turbulence, which are simply a relabelling of text book equations from quantum field theory with variables appropriate to turbulence. Yet such theories do not engage with the existing body of work or explain how they solve problems that others encountered. They used to appear in obscure journals of the old Soviet Union, but now they appear in the learned journals of the west. It appears that the authors do not understand that their work is unsound or perhaps do not care. I intend to write on the subject of Fake Theories (don’t know what put that idea in my head!) but as a topic it presents its difficulties.

Lastly, for completeness, I should mention that there is a class of theories based on the use of Lagrangian coordinates. A recent development in this type of theory also presents a decent and balanced review of other work in the field [4]. I also intend to write about Lagrangian theories in a future post.

[1] W. D. McComb. Dynamics and Relativity. Oxford University Press, 1999.
[2] W. D. McComb and S. R. Yoffe. A formal derivation of the local energy transfer (LET) theory of homogeneous turbulence. J. Phys. A: Math. Theor., 50:375501, 2017.
[3] P. G. Saffman. Problems and progress in the theory of turbulence. In H. Fiedler, editor, Structure and Mechanisms of Turbulence II, volume 76 of Lecture Notes in Physics, pages 273-306. Springer-Verlag, 1977.
[4] Makoto Okamura. Closure model for homogeneous isotropic turbulence in the Lagrangian specification of the flow field. J. Fluid Mech., 841:133, 2018.

Turbulent dissipation and the two cultures?
I recently saw the paper cited as [1] below, which for me is I think the first of the 2021 papers. As the title suggests, it presents a review of methods of measuring the turbulent dissipation rate. It contains a certain amount of basic theory, along the lines of expressions for the dissipation rate, the Taylor dissipation surrogate, remarks about the role of inertial transfer, dynamical equilibrium, and so on. But there is no attempt at a statistical theory and most theoretical attempts at explaining the dependence of the dissipation rate $\varepsilon$ on the Taylor-Reynolds number do not get a mention.

Nevertheless, the authors do cite a paper by my co-authors and me, which presents an analytical theory of the dependence of the normalized dissipation $C_{\varepsilon}$, which is the dissipation rate divided by $U^3/L$, where $U$ is the root mean square velocity and $L$ is the integral lengthscale [2]. They say that we `explained that the decay of the dimensionless dissipation with increasing Reynolds number was because of the increase in the Taylor surrogate’. This is true for forced, stationary turbulence, because we can keep the rate of forcing (and hence the dissipation) constant while decreasing the viscosity in order to increase the Reynolds number.

However, this paper says so much more! It presents an analytical theory, based on the Karman-Howarth equation, in which dimensionless structure functions are expanded in inverse powers of the Reynolds number. The resulting expression is given by: \begin{equation}C_{\varepsilon}= C_{\varepsilon,\infty}+C/R_L + O(1/R^2_L), \end{equation} where $R_L$ is the integral scale Reynolds number. Direct numerical simulation was used to obtain the coefficients as $C=18.9 \pm 0.009$ and $C_{\varepsilon,\infty}= 0.468\pm0.006$. The result compared to other numerical investigations is shown in the figure below, which is taken from Fig. 1 of [2], and where equation (44) of that reference is the equation just given here.

It is worth emphasising that this result is asymptotically exact in the limit of large Reynolds numbers. For low Reynolds numbers, our DNS confirmed that the $1/R_L$ dependence was correct to within experimental error. When this theory was later applied to magnetohydrodynamic turbulence, it was found necessary to include a term at order $1/R^2_L$ at low Reynolds numbers [3]. In fact, a detailed argument was previously put forward by us to the effect that the $1/R_L$ dependence was exact for isotropic turbulence: see the supplemental material to the paper cited as [4] below.

I should also emphasise that none of this is intended as a criticism of Wang et al, which is a perfectly competent piece of work of its general type. It is really a matter of emphasising the gulf between fluid dynamics and physics. For instance, it would be very unlikely that an experimental particle physicist would fail to see the point of a paper by a theoretical particle physicist, even if they were unable to follow the detailed derivations in it. This is because in physics we all have the same education up to a certain level, and even thereafter there is overlap and much in common. But fluid dynamics is much less homogeneous than physics and this leads to misunderstandings based very largely on cultural gaps. Those of us who belong to the very small number of physicists working on turbulence have much cause to be aware of this. I have posted about this before and I will do so soon again!

[1] Guichao Wang et al. Estimation of the dissipation rate of turbulent kinetic energy: A review. Chem. Eng. Science, 229:11633, 2021.
[2] W. D. McComb, A. Berera, S. R. Yoffe, and M. F. Linkmann. Energy transfer and dissipation in forced isotropic turbulence. Phys. Rev. E, 91:043013, 2015.
[3] M. F. Linkmann, A. Berera, W. D. McComb, and M. E. McKay. Nonuniversality and Finite Dissipation in Decaying Magnetohydrodynamic Turbulence. Phys. Rev. Lett., 114:235001, 2015.
[4] W. David McComb, Arjun Berera, Matthew Salewski, and Sam R. Yoffe. Taylor’s (1935) dissipation surrogate reinterpreted. Phys. Fluids, 22:61704, 2010.