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.
Principal Thermal Conductivity and Rotating Angle Measurement of Two-Dimensional Anisotropic Heat Conductor Using an Unsteady Point Source of Heat.