Even linear stability problems in fluid dynamics can be fiendishly difficult if the basic state has some kind of non-uniform spatial structure, so that the differential equations have variable coefficients. In the late 1970s and early 1980s I set about developing a purely local approach to a variety of different instabilities, assuming that disturbances have very small wavelength in the direction of the non-uniformity.
This resulted, for example, in clarification of the precise conditions for Goldreich-Schubert instability of differential rotation, and a simplification and extension of R.J. Tayler's conditions for magnetic instabilities in stars. It also showed how 3 rather disturbingly different studies of magnetic buoyancy (Gilman 1970, Roberts & Stewartson 1977, Acheson & Gibbons 1977) were consistent with one another, each being valid in an appropriate region of parameter space.
The whole approach met with some scepticism at the time, particularly for its complete neglect of boundary conditions, but subsequent research has confirmed, I believe, that these methods can successfully capture the essence of quite complicated stability problems. In particular, they can help identify the key dimensionless parameters on which the stability of a system really depends. Nonetheless, which kind of problems yield well to this technique and which do not is still rather an open question.
Phil. Trans. Roy. Soc. Vol 289, pp 459-500, 1978
J. Fluid Mech. Vol 85, pp 743-757, 1978 (with M.P. Gibbons)
Geophys. Astrophys. Fluid Dyn. Vol 27, 123-136, 1983
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