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1. Moffatt, H.K. 1961 The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity.  J. Fluid Mech11, 625-635.


This was my first paper, published while I was still a research student under the supervision of G.K.Batchelor FRS, already famous by 1961 as Founder-Editor of the Journal of Fluid Mechanics as much as for his fundamental contributions to the theory of turbulence.  With his encouragement, I built on his earlier (1959) work on the action of turbulence on a passive scalar field, applying the same ideas to a passive vector field, specifically the weak magnetic field of the title of this paper.  Among the results was a prediction that the spectrum of magnetic fluctuations should fall off like k^{-11/3}, in agreement with a result obtained independently one year earlier by G.S. (Giorgy) Golitsyn at  the Institute of Atmospheric Physics in Moscow.


This prediction was verified many years later  by Odier, Pinton & Fauve (1998) who measured the magnetic spectrum in a turbulent flow of liquid gallium in the presence of a weak applied magnetic field.  There is great satisfaction for an applied mathematician in having a theoretical prediction  experimentally confirmed in this way, the more so if the waiting time is long!


Batchelor, G.K., Howells, I.D. & Townsend, A.A. 1959, J.Fluid Mech., 5, 134-139.

Golitsyn, G.S. 1960, Soviet Phys. Doklady, 5, 536.

Odier,P., Pinton, J.-F. & Fauve, S. 1998 Phys. Rev.E, 58, 7397--7401.

In September 1961, a famous  International Colloquium was held in Marseille on the subject of turbulence. Many eminent scientists were present, including Theodore von KarmanGeoffrey Ingram Taylor and Andrei Nicolaevich Kolmogorov. I was honoured to lecture on the subject “Turbulence in Conducting Fluids”, a baptism of fire in the face of such an audience, from which I was lucky to emerge unscathed.

Here is one of the few surviving photos taken at the Colloquium banquet, Marseille 1961. Leslie Kovasznay is at the head of the table; G.I.Taylor on his left; George Batchelor in white shirt with his back to the camera, in conversation with Anna Kovasznay; Mark Morkovin, Don Coles, Itiro Tani opposite.  

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I have written about this historic event more recently in: 

Moffatt, H.K. 2012 Homogeneous turbulence; an introductory review. Journal of Turbulence13, 1-11.

7. Moffatt, H.K. 1964 Viscous and resistive eddies near a sharp corner.  J.Fluid Mech18, 1-18

1960 2.jpg

In considering flow near a sharp corner between two bounding planes, it is natural to look for a solution of the biharmonic equation proportional to some power λ of distance from the corner.  The big surprise in this work was the fact that λ turns out to be a complex number when the angle of the corner is less than about 146º, implying infinite oscillations, i.e. an infinite sequence of counter-rotating eddies as the corner is approached.   These eddies decrease very rapidly in intensity and are difficult to observe experimentally. 

However, here is the photo that Sadatoshi Taneda (Kyushu University, Japan) sent me in 1979, showing the flow in a corner driven by rotation of a remote cylinder at Reynolds number 0.17, providing brilliant confirmation of the validity of the theory, even although only two of the infinite sequence of eddies are apparent; even this required a very long time-exposure (~ 90 minutes).

I was led to this problem through giving a graduate (‘Part III) course in Cambridge on the subject “Slow Viscous Flow” and being obliged to think of suitably teasing problems for the examination at the end of the year.  This problem turned out to be a bit too teasing!   It transpires that these corner eddies are a generic phenomenon that appears in a great variety of flow situations. Incidentally, the ‘resistive’ eddies that are also analysed in this paper are the magnetohydrodynamic counterpart of the ‘viscous’ eddies.


For me, teaching at both undergraduate and graduate level has frequently led to fascinating research problems of this kind.   I am a firm believer in the fruitful interaction between teaching and research activity.  Teaching demands a certain breadth of outlook;  and the need to explain things in the simplest and clearest way can often lead to new research insights.


I presented this work at a Symposium held in Zakopane, Poland, 1963, hosted by Władek Fiszdon.  At this meeting, I met Milton van Dyke, who invited me to spend my first sabbatical leave at Stanford University (March-July 1965)

Taneda, S. 1979 Visualization of separating Stokes flows. J. Phys. Soc. Japan, 46, 1935-1942.

see also

Branicki, M. & Moffatt, H.K. 2006 Evolving eddy structures in oscillatory Stokes flows in domains with sharp corners. J. Fluid Mech., 551, 63-92.

10. Moffatt, H.K. 1967 Interaction of turbulence with strong wind shear. In Atmospheric Turbulence and Radio Wave Propagation (Eds. A.M.Yaglom & V.I.Tatarski), Nauka, Moscow, 139-156.

This paper, presented at the International Colloquium on Atmospheric Turbulence and Radio Wave Propagation (Moscow, July 1965), shows what is now recognised as the ‘transient instability’ of a uniform shearing flow, limited only by the effect of weak viscosity.  The lowest curve here (from the paper) shows this transient linear growth of the streamwise component of a typical disturbance to the flow. The same behaviour was later found quite independently by Landahl (1980), and has led to many significant developments.

I spent the period March - July 1965 as a Visiting Assistant Professor at the Department of Aero- and Astronautics, Stanford University, where I taught a graduate course on Turbulence. In struggling to understand Townsend’s ‘large eddy’ theory as presented in his 1956 book, I was led to analyse the shearing of homogeneous turbulence by ‘rapid distortion theory’. The Moscow meeting provided an ideal opportunity to present this work.

1960 3.png

Landahl, M.T. 1980  A note on an algebraic instability of inviscid parallel shear flows. J.Fluid Mech. 98, 243-251.

Townsend, A.A. 1956 The Structure of Turbulent Shear Flow. Cambridge Univ. Press.

14. Moffatt, H.K. 1969 The degree of knottedness of tangled vortex lines.  J.Fluid Mech35, 117-129.


In 1868, Lord Kelvin (then W.Thomson) recognised that vortex lines are ‘frozen’ in any inviscid barotropic flow under conservative body forces, and that vortex tube topology must therefore be conserved.  In my 1969 paper, I introduced an integral quantity, the helicity of the flow, and showed that this is an invariant of the Euler equations, precisely because of this conservation of topology.


I came upon this result through struggling to find a physical interpretation of a result of L. Woltjer (1958) who had shown the invariance of a quantity, now known as ‘magnetic helicity’, in a perfectly conducting fluid.  I was giving a Part III course on Magnetodynamics of Fluids  in 1967, and it suddenly dawned on me that Woltjer’s result followed simply from the fact that the lines of force of the magnetic field are frozen in the fluid. It followed that there must be an analogous result for the nonlinear Euler equations reflecting the fact that vortex lines are similarly ‘frozen-in’ in this context.  The rest was straightforward!  It is remarkable that the invariance of helicity, though simple to prove, had remained hidden for close on 100 years since the discoveries of Helmholtz and Kelvin.

1960 4.png

There was a precursor to my paper in a short paper of  R. Betchov (1961) who recognised the potential importance of helicity, but had no idea of its Euler-invariance.  This invariance was in fact proved by J.-J. Moreau (1961).  I became aware of the existence of these two papers only much later. It so often happens that results we publish turn out to have been proved much earlier,  although their true significance may have escaped attention (as did Moreau’s paper until it came to my attention in 1979; I cited it in 1981).

Betchov, R. 1961  Semi-isotropic turbulence and helicoidal flows. Phys. Fluids, 4, 925-926.

Moffatt, H.K. 1981 Some developments in the theory of turbulence.  J.Fluid Mech. 106, 27-47.

Moreau, J.-J. 1961 Constantes d’un îlot tourbillonnaire en régime permanent. C.R. Acad. Sci. Paris, 252, 2810-2813.

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