LINEAR AND NONLINEAR DYNAMICS OF A DIFFERENTIALLY HEATED SLOT UNDER GRAVITY MODULATION

In this paper we consider the effect of sinusoidal gravity modulation of size epsilon on a differentially heated infinite slot in which a ve rtical temperature stratification is imposed on the walls. The slot pr oblem is characterized by a Rayleigh number, Prandtl number, and the i mposed uniform stratification on the walls. When epsilon is small, we show by regular perturbation expansion in epsilon that the modulation interacts with the natural mode of the system to produce resonances, c onfirming the results of Farooq & Homsy (1994). For epsilon similar to O(1) we show that the modulation can potentially destabilize the long wave eigenmodes of the slot problem. This is achieved by projecting th e governing equations onto the least-damped eigenmode, and investigati ng the resulting Mathieu equation via Floquet theory. No instability w as found at large values of the Prandtl number and also low stratifica tion, when there are no travelling modes present. To understand the no nlinear saturation mechanisms of this growth, we consider a two-mode m odel of the slot problem with the primary mode being the least-damped travelling parallel-flow mode as before and a secondary mode of finite wavenumber. By projecting the governing equations onto these two mode s we obtained the equations for temporal evolution of the two modes. F or modulation amplitudes above critical, the growth of the primary mod e is saturated resulting in a stable weak nonlinear synchronous oscill ation of the primary mode. An unexpected and intriguing feature of the coupling is that the secondary mode exhibits very high-frequency burs ts which appear once every cycle of the forcing frequency.