Some developments in the theory of turbulence

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Abstract
This is in no way intended as a review of turbulence-the subject is far too big for adequate treatment within a reasonably finite number of pages; the monumental treatise of Monin & Yaglom (1971, 1975) bears witness to this statement. It is rather a discourse on those aspects of the problem of turbulence with which I have myself had contact over the last twenty years or so. My choice of topics therefore has a very personal bias - but that is perhaps consistent with the style and objectives of this rather unusual issue of JFM.I have approached the dynamical problem of turbulence via two simpler (but nevertheless far from trivial) problems – viz the convection and diffusion of a passive scalar field and of a passive vector field by turbulence of known statistical properties. Particular emphasis is given to the method of successive averaging (a simplified version of the renormalization-group technique) which seems to me to have considerable potential. The difficulty of extending this method to the dynamical problem is discussed.In a final section (§ 6) I have allowed myself the luxury of discussing a somewhat esoteric topic - the problem of inviscid invariants and their relationship with the topological structure of a complex vorticity field. The helicity invariant is the prototype; it is identifiable with the Hopf invariant (1931) and it may be obtained from appropriate manipulation of Noether's theorem (Moreau 1977). A suggestion is made concerning possible measurement of this fundamental measure of ‘lack of reflexional symmetry’ in a turbulent flow.

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