7 - MHD Turbulence at Low and High Magnetic Reynolds Number

Summary

You asked, ‘What is this transient pattern?’ If we tell the truth of it, it will be a long story; It is a pattern that came up out of an ocean And in a moment returned to that ocean's depth

(Omar Khayyam)

Turbulence is not an easy subject. Our understanding of it is limited, and those bits we do understand are arrived at through detailed and difficult calculation. G K Batchelor gave some hint of the difficulties when, in 1953, he wrote:

It seems that the surge of progress which began immediately after the war has now largely spent itself, and there are signs of a temporary dearth of new ideas… we have got down to the bedrock difficulty of solving non-linear partial differential equations.

Little has changed since 1953. Nevertheless, it is hard to avoid the subject of turbulence in MHD, since the Reynolds number, even in metallurgical MHD, is invariably very high. So at some point we simply have to bite the bullet and do what we can. This chapter is intended as an introduction to the subject, providing a springboard for those who wish to take it up seriously. In order not to demotivate the novice, we have tried to keep the mathematical difficulties to a minimum. Consequently, only schematic outlines are given of certain standard derivations and proofs. For example, deriving the standard form for second- and third-order velocity correlation tensors in isotropic turbulence can be hard work. Such derivations are well documented elsewhere and so there seems little point in giving a blow-by-blow description here.