Taylor-Goldstein equation and stability

Taylor-Goldstein equation (TGE) governs the stability of a shear-flow of an inviscid fluid of variable density. It is investigated here from a rigorous geometrical point of view using a canonical class of its transformations. Rayleigh's point of inflection criterion and Fjortoft's condition of instability of a homogenous shear-flow have been generalized here so that only the profile carrying the point of inflection is modified by the variation of density. This fulfils a persistent expectation in the literature. A pair of bounds exists such that in any unstable flow the flow-curvature (a function of flow-layers) exceeds the upper bound at some flow-layer and falls below the lower bound at a higher layer. This is the main result proved here. Bounds are obtained on the growth rate and the wave numbers of unstable modes, in fulfillment of longstanding predictions of Howard. A result of Drazin and Howard on the boundedness of the wave numbers is generalized to TGE. The results above hold if the local Richardson number does not exceed 1/4 anywhere in the flow, otherwise a weakening of the conditions necessary for instability is seen. Conditions for the propagation of neutrally stable waves and bounds on the phase speeds of destabilizing waves are obtained. It is also shown that the set of complex wave velocities of normal modes of an arbitrary flow is bounded. Fundamental solutions of TGE are obtained and their smoothness is examined. Finally sufficient conditions for instability are suggested.