Transitional regimes of penetrative convection in a plane layer

Abstract

Hydrodynamical regimes are described for penetrative convection in a two-dimensional bounded plane layer of water at the temperature range close to the point of maximum density. Stress-free conditions on the horizontal and vertical boundaries of the domain are assumed. The point of maximum density is supposed to be located in the middle plane of the layer in conductive state. Steady and time-periodic solutions are found on large horizontal scales and the lengths of a spatial periodicity inside the layer are determined. Hydrodynamical peculiarities of time-periodic solution are described. In the domain equal to the periodicity cell corresponding to time-periodic solution, the sequence of hydrodynamical regimes with the increase of supercriticality is analyzed and the full sequence of bifurcations from conductive state to chaotic motion is described. The time-periodic solution loses the stability through the subcritical Neimark–Sacker bifurcation. First, the closed invariant curve has one stable period-2 cycle, so the bifurcation looks like period doubling. With a further increase of supercriticality the period-2 cycle transforms to a torus through the saddle-node bifurcation for maps. Then, loss of stability of quasiperiodic motion occurs through intermittency and chaotic mixing on the background of quasiperiodic solution. The window of quasiperiodicity was obtained in the interval of supercriticality corresponding to intermittent regimes, after which intermittency on the background of another quasiperiodic mode appears.