The Rayleigh-taylor Instability of Transition Layer

New types of symmetry for the Rayleigh equation are found. For small Atwood number, an analytic solution is obtained for a smoothly varying density profile. It is shown that a transition layer with a finite width can undergo some kind of stratification. PACS numbers: 47.20.-k, 52.35.Py. e-mail:[email protected] Although Rayleigh had investigated the instability (known now as Rayleigh-Taylor (RT) instability) using an exponential profile of density [1], most investigations have been done so using sharp interfaces [2-6]. In [7] it is carried out making an interesting analogy between the equations which describes the RT instability, and the Schrodinger equation. In that study, an “equivalent potential” is constructed and for well-known density profiles it is shown that the Schrodinger equation with the corresponding “equivalent potential” has the same eigenvalues as the Rayleigh equation. The physical quantities (e.g.; density, velocity) and their derivatives, generally speaking cannot suffer a jump discontinuity. Therefore, it seems more consistent to attempt to solve the problem for a transition layer of finite thickness and then take the limit when the thickness of this layer ∆ tends to zero; i.e. consider the case of a density jump in the limit, ∆ → 0. The solution for this transition layer gives us a more complete physical picture of the instability. In this paper we consider the RT instability of a transition layer of finite thickness, where the unperturbed density changes continuously from a constant value up to another one. For that, an analytic solution is found, in the limit of small Atwood number. This solution allows us to investigate the ∆ → 0 limit. We find that: 1. For an arbitrary density profile the “equivalent potential” can be presented in compact form. This allowed us to predict a new type of symmetry (in addition to the well-known symmetry of the Rayleigh equation [2]). 2. The depth of the “equivalent potential” well depends on the width of the transition layer and the depth not the energy of the potential well undergoes a quantization. For the finite width of the layer the spectrum of eigenvalues is infinite. This conclusion is consistent with the fact that with the increase of the quantum number grows the well depth. 3. The eigenfunctions found here show the exfoliation of the transition layer. I. Using the linearized equation of motion and the continuity equation with the help of