Big whirls have little whirls
that feed on their velocity,
and little whirls have lesser whirls
and so on to viscosity.
It takes completely different physics to model whitewater than flatwater, but why does this transition happen so suddenly, as when an ocean wave breaks into foam? In general, slow moving viscous fluids are extremely well understood, using laminar fluid dynamics which were developed over the course of centuries.
Turbulence, on the other hand, has been understood at about the level of the opening poem for centuries. Turbulent “whirls” are called eddies, or eddy currents, and when Arago and Foucault discovered similarly shaped electrical currents in conductors they assumed they had a similar cause. But, as Maxwell made clear, they had nothing to do with turbulent eddies. Rather it was quite the opposite. Steady current in normal conductors follows the local electric field according to Ohm’s law. Superconductors, on the other hand, act as fully turbulent electrical fluids. As such, they were one of the most remarkable discoveries of the early twentieth century.
In 1911, Heike Kamerlingh Onnes first observed that, at temperatures near absolute zero, the resistivity of liquid helium vanished.[3] In 1933 Walther Meissner and Robert Ochsenfeld found that magnetic fields do not penetrate superconductors, in much the same way as the electric field goes to zero inside a normal conductor. This was confusing at the time, as is nicely outlined in the opening address to a 1935 conference on the topic by the Canadian physicist John Cunningham McLennan.
The series of experiments on magnetic fields around supraconductors, commenced by Meissner and his co-workers, has recently opened a new method of attack on this problem. … Meissner and his collaborators in Berlin began by studying the field distribution round cylinders, and inside a hollow cylinder, with a small test coil. …
In agreement with simple electromagnetic considerations, the supraconductor acts as if its magnetic permeability were zero, at any rate beyond a very thin surface layer. But when the transition to the supraconducting state takes place in the presence of a magnetic field, either by cooling in a steady external field or by reducing the field strength from an initial value greater than the critical field, the experiments agree only in that the result is never quite what was expected. However, certain experimental facts seem to stand out, at least for most metals which have been tested. First, the distribution of external field is spontaneously readjusted nearly to the distribution it would have if the induction within the body were everywhere zero. Second, the field inside a hollow is only slightly altered, if at all, in the transition, showing the curious phenomenon of an apparently isolated section of magnetic field in a stable state. Third, for a solid body varying amounts of the original magnetic flux are “locked in” in the course of the transition. The field thus locked in, like the field inside the hollow, cannot then be disturbed by any external influence so long as the body remains in the supraconducting state.[4]
Notice that McLennan uses the word magnetic field to mean the magnetic field applied externally in the laboratory, H, and the word induction to mean the magnetic field, B, inside the “supraconductor,” as in the Heaviside convention.
These experimental results led the brothers, Fritz and Heinz London (also in attendance), to come up with a classical model, which they referred to as acceleration theory, to replace Ohm’s law.[5]
As the opening poem states, when stirred on large scales, “big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls.” This cascade transfers kinetic energy from slowly moving large eddies to rapidly moving tiny eddies that dissipate the energy—the lower the viscosity the smaller the tiny eddies.
What happens when the electron sea has absolutely no viscosity? The faster the electrons move, J, the greater the sideways magnetic force they encounter. On the other hand, the smaller the whirl, the curl of J, the stronger the magnetic field they induce. As is usually the case in a stochastic processes, these energies quickly even out. In any given volume, half of the energy resides in the magnetic field and the other half in the electrons. Thus, in superconductors, if current is whirling, there must be a magnetic field. Similarly, if there is no magnetic field, there must also be no current.
We commence the chapter with a review of viscosity and a discussion of turbulent fluid flow. This leads us to London acceleration theory and how it explains the odd properties of superconductors. [6]
[1] Richardson, Lewis F. Weather Prediction by Numerical Process. 1st ed. (Cambridge University Press, 1922), retrieved from https://archive.org/.
[2] This little ditty was based on a nursery rhyme, which, in-turn, came from a line in Jonathan Swift’s 1733 poem On Poetry: a Rhapsody that reads: “So, naturalists observe, a flea / Has smaller fleas that on him prey / And these have smaller still to bite ’em, / And so proceed ad infinitum.”
[3] The top picture is of Paul Ehrenfest, Henrick Lorentz, Neils Bohr, and Kamerlingh Onnes. The bottom is a diagram from Onnes’s 1913 Nobel lecture.
[4] J.C. McLennan, et al., “A Discussion on Supraconductivity and Other Low Temperature Phenomena,” Proceedings of the Royal Society of London A, 152 (1935), 1-46.
[5] F. London, and H. London, “The Electromagnetic Equations of the Supraconductor,” Proceedings of the Royal Society of London A, 149 (1935), 71-88.
[6] For further discussion of the behavior of a superconductor in a magnetic field, see, for example, K.B. Ma, et al., “Superconductor and magnet levitation devices,” Review of Scientific Instruments 74, no. 12, (2003), 4990-5017.
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