Finite-amplitude perturbations of the advective flows in a horizontal layer of incompressible liquid with a free upper boundary at low rotation

Abstract

The behavior of spatially finite-amplitude perturbations of the advective flow in a rotating horizontal layer of incompressible liquid with a free upper boundary and a lower rigid boundary at low rotation is investigated. The study is based on the equations of convection in the Boussinesq approximation in a rotating reference frame in a Cartesian coordinate system. Because of the great complexity of a three-dimensional formulation the following limiting cases are considered: spatial perturbations of the first type in the form of rolls with axes perpendicular to x -direction and spatial perturbations of the second type in the form of rolls with axes parallel to x -axis. At a fixed Prandtl number (Pr=6.7), perturbation isolines are plotted for different Taylor and Grashof numbers. The behavior of finite-amplitude perturbations is considered beyond the stability threshold. The nonlinear problem is solved numerically by the grid method. An explicit finite-difference scheme with central differences is used. The Poisson equation for perturbation stream functions is solved by the successive over-relaxation (SOR) method. The numerical results allow one to evaluate the behavior of perturbations and to determine velocity characteristics, amplitude and period of repetition of perturbations. The analysis indicates that spatial structures (vortices) oriented across the layer are generated in the supercritical region under the influence of temperature inhomogeneities. Temperature perturbation is a system of alternating warm and cold spots located along the direction of the temperature gradient on the layer boundaries. With increasing Taylor number, the vortices near the free upper boundary are replaced by those along the lower rigid boundary, and the amplitude of vortices increases. With increasing Grashof number, thermal spots are rearranged, their size increases, the warm spots are located near the upper and lower boundaries of the layer, and the cold ones in the center of the layer. The movement becomes more complex. A repetition period of finite amplitude perturbation patterns reduces.