*`Local' methods for determining
instability*

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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