THE TILTING INSTABILITY WITH BUOYANT FORCING IN A 2-DIMENSIONAL VISCOUS-FLUID

The tilting instability is an instability of a two-dimensional fluid t hat transforms convective motion into shear flow. As a generalization of previous analytical work on the tilting instability in an ideal flu id, the authors investigate the instability with thermal buoyancy incl uded as a source supporting convection against viscous dissipation; Th e results show two distinct instabilities: for large Rayleigh numbers, the instability is similar to the tilting instability in an inviscid fluid; for small Rayleigh numbers, it resembles a dissipative (i.e., v iscous) instability driven by thermal buoyancy. This paper presents a linear stability analysis together with numerical solutions describing the nonlinear evolution of the flow for both types of instabilities. It is shown that the tilting instability develops for values of the as pect ratio (the ratio of the horizontal spatial scale to the vertical scale) that are less than unity. In the case of an ideal fluid, the in stability completely transforms the convection into a shear flow, whil e the final stage of the dissipative instability is one of coexisting states of convection and horizontal shear flow. This study is confined to two dimensions, and the role of the tilling instability in three d imensions remains a subject for future research. In two dimensions, ho wever, the tilting instability can readily generate shear flows from c onvective motions, and this mechanism may well be important in the int erpretation of the results of two-dimensional numerical simulations.