INSTABILITY OF STRATIFIED FLUID IN A VERTICAL CYLINDER

In a previous paper we analysed the stability to small disturbances of stationary stratified fluid which is unbounded. Various forms of the undisturbed density distribution were considered, including a sinusoid al profile and a function of the vertical coordinate z which is consta nt outside a central horizontal layer. Both these types of stratificat ion are so unstable that the critical Rayleigh number is zero. In this sequel we make the study more complete and more useful by taking acco unt of the effect of a vertical circular cylindrical boundary of radiu s a which is rigid and impermeable. As in the previous paper we assume that the undisturbed density distribution is steady. The case of flui d in a vertical tube with a uniform density gradient is useful for com parison, and so we review and extend the available results, in particu lar obtaining growth rates for a disturbance which is neither z-indepe ndent nor axisymmetric. A numerical finite-difference method is then d eveloped for the case in which d(rho)/d(z) = rho0 kappaA COS kappaZ. W hen kappaa much less than 1 the relation between growth rate and Rayle igh number approximates to that for a uniform density gradient of magn itude pho0 kappaA; and when kappaa much greater than 1 the tilting-sli ding mechanism identified in the previous paper is relevant and the re sults approximate to those for an unbounded fluid, except that the sma llest Rayleigh number for a neutral disturbance is not zero but is of order (kappaa)-1. In the case of an undisturbed density which varies o nly in a central layer of thickness l, the same mechanism is at work w hen the horizontal lengthscale of the disturbance is large compared wi th l, resulting in high growth rates and a critical Rayleigh number wh ich vanishes with l/a. Estimates of the growth rate are given for some particular density profiles.