Thermal convection under external modulation of the driving force. I. The Lorenz model.

Abstract

Thermal convection between horizontal plates is considered for a situation in which the driving force varies periodically in time. This variation may come from changes in the temperature or the plates or from a vertical oscillation of the cell, causing variation of the gravitational force. Truncation of the Boussinesq equations leads to a three-mode model which is a generalization of the Lorenz model to the case of external modulation. Similar models have previously been introduced by Finucane and Kelly for stress-free horizontal boundaries and by Gresho and Sani for the rigid-boundary case. The threshold behavior is that of a parametrically driven damped oscillator, whose bifurcations are studied numerically, as well as analytically in certain limits. It is found that in general the modulation stabilizes the conducting state. For stress-free horizontal boundaries the threshold shifts predicted by the model coincide with the results of Rosenblat and Herbert in the limit of low frequency and agree well with Venezian's results for small modulation amplitude, both obtained using the full Boussinesq equations. For rigid boundaries the results agree well with numerical calculations of Rosenblat and Tanaka. The nonlinear behavior of the model is also studied, and the convective contribution to the heat current evaluated. The Lorenz model is shown to reproduce, either exactly or to a good approximation, most previous theoretical results on modulated convection, and the model can be studied simply for a wide range of parameters. The above discussion refers to a laterally infinite system. For a real finite system, sidewall effects are shown to cause a rounding of the convective threshold in the presence of modulation, particularly at low frequencies. A calculation of these effects is carried out within the framework of the Lorenz truncation, and the resulting imperfect bifurcation of the model is studied numerically. In a companion paper (immediately following this one) quantitative experimental results are presented and compared to the predictions of the Lorenz model.