Secondary bifurcation in convective shells.

Abstract

A fluid-filled horizontal annulus is formed by two coaxial circular cylinders of nearly equal radii. When subjected to heating, there is one convective state that is uniform in the axial coordinate and so is equivalent to the steady loop convection of the Lorenz equations. This loop convection is susceptible to axially periodic instabilities analogous to the zigzag or cross-roll instabilities on two-dimensional B\'enard convection. The loop convection bifurcates from the conductive state at Rayleigh number R=1 and is itself unstable to axially periodic steady convection at a critical value R=${R}_{s}$(k) at a wave number k where ${R}_{s}$\ensuremath{\rightarrow}1 as k\ensuremath{\rightarrow}0 so that asymptotically the loop solution is always unstable to steady long-wave instabilities. We analyze the long-wave structure by deriving a coupled system of nonlinear evolution equations that give the interaction of the k=0 loop solution and the k\ensuremath{\rightarrow}0 spatially periodic convection.