Convective instabilities in a rotating vertical Hele-Shaw cell

Convective instabilities driven by vertical buoyancy in a Boussinesq fluid in a rotating vertical Hele-Shaw cell, a long channel with rectangular cross section of finite height h and small width Gamma h with Gamma < 1, are investigated both analytically and numerically. The problem is characterized by the Taylor number T, the Rayleigh number R, and the aspect ratio Gamma. Explicit asymptotic solutions describing convective instabilities are derived for Gamma T-1/6 < 1, where T is assumed to be large compared to unity. Comparison between the asymptotic and fully numerical solutions shows a satisfactory quantitative agreement. It is found that an overall condition for convective instabilities becomes optimal when Gamma T-1/6=O(1). Direct three-dimensional simulations for strongly nonlinear convection are also carried out in the regime 0 <(R-R-c)/R-c <= O(10), where R-c denotes the critical Rayleigh number. As a consequence of both the geometric and dynamic constraints imposed by the narrow channel (geometric) and rapid rotation (dynamic), the nonlinear flow remains temporally stationary and spatially simple and is comprised primarily of vertically long thin convection cells that transport heat all the way from the bottom to the top of the channel. (c) 2006 American Institute of Physics