Cellular structure in a natural convection loop and its chaotic behaviour: II. Theory

A theoretical study of thermal convection in a vertically oriented thin toroidal loop which is placed in a uniform negative vertical temperature gradient is reported. The Boussinesq approximation is employed, and the first-order perturbed fields from a steady conduction state are obtained in the form of a double Fourier series. The critical Rayleigh number for the onset of convection is determined and various types of steady convective patterns, including cellular structure, are examined. Four typical modes are superposed to express the time-dependent velocity and temperature fields, whose mode-amplitudes constitute an extended version of the Lorenz model. Numerical simulation shows a sequence of transitions from steady to periodic, quasi-periodic, and chaotic states as the Rayleigh number is increased. The present model successfully explains our experimental results.