Generalized Energies and the Lyapunov Method
Thus far in convection studies we have explored uses of the energy method that have concentrated on employing some form of kinetic-like energy, involving combinations of L 2 integrals of perturbation quantities. While this is fine and yields strong results for a large class of problem, there are many situations where such an approach leads only to weak results if it works at all, and an alternative device must be sought. In fact, recent work has often employed a variety of integrals rather than just the squares of velocity or temperature perturbations. Drazin & Reid (1981), p. 431, point out that this natural extension of the energy method is essentially the method advocated by Lyapunov for the stability of systems of ordinary differential equations some 60 years or so ago. In this chapter we indicate where a variety of different generalized energies (or Lyapunov functionals) have been employed to achieve several different effects. Since the applications are usually connected to geophysical or astrophysical problems we discuss this aspect also: chapter 7 is, however, devoted entirely to two geophysical problems where energy theory has proved valuable. One of these, convection in thawing subsea permafrost, is particularly attractive from an energy stability point of view because the construction of a novel generalized energy is necessary to achieve sharp quantitative stability bounds.
KeywordsRayleigh Number Generalize Energy Stability Boundary Critical Rayleigh Number Taylor Number
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