Generation of collimated jets by a point source of heat and gravity

Abstract

New solutions of the Boussinesq equations describe the onset of convection as well as the development of collimated bipolar jets near a point source of both heat and gravity. Stability, bifurcation, and asymptotic analyses of these solutions reveal details of jet formation. Convection (with l cells) evolves from the rest state at the Rayleigh number Ra = Racr = (l − 1)l(l + 1)(l + 2). Bipolar (l = 2) flow emerges at Ra = 24 via a transcritical bifurcation: Re = 7(24 − Ra)/(6 + 4Pr), where Re is a convection amplitude (dimensionless velocity on the symmetry axis) and Pr is the Prandtl number. This flow is unstable for small positive values of Re but becomes stable as Re exceeds some threshold value. The high-Re stable flow emerges from the rest state and returns to the rest state via hysteretic transitions with changing Ra. Stable convection attains high speeds for small Pr (typical of electrically conducting media, e.g. in cosmic jets). Convection saturates due to negative ‘feedback’: the flow mixes hot and cold fluids thus decreasing the buoyancy force that drives the flow. This ‘feedback’ weakens with decreasing Pr, resulting in the development of high-speed convection with a collimated jet on the axis. If swirl is imposed on the equatorial plane, the jet velocity decreases. With increasing swirl, the jet becomes annular and then develops flow reversal on the axis. Transforming the stability problem of this strongly non-parallel flow to ordinary differential equations, we find that the jet is stable and derive an amplitude equation governing the hysteretic transitions between steady states. The results obtained are discussed in the context of geophysical and astrophysical flows.