The tilting instability with buoyant forcing in a two-dimensional viscous fluid

The tilting instability is an instability of a two-dimensional fluid that transforms convective motion into shear flow. As a generalization of previous analytical work on the tilting instability in an ideal fluid, the authors investigate the instability with thermal buoyancy included as a source supporting convection against viscous dissipation; The 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., viscous) 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 aspect 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 instability completely transforms the convection into a shear flow, while 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 dimensions remains a subject for future research. In two dimensions, however, the tilting instability can readily generate shear flows from convective motions, and this mechanism may well be important in the interpretation of the results of two-dimensional numerical simulations.