The route to chaos in thermal convection at infinite Prandtl number: 1. Some trajectories and bifurcations

Abstract

An important issue is whether thermal convection in the Earth's mantle is chaotic and the mechanisms by which this chaos may be generated. In this paper, we conclude that the high Prandtl number/high Raleigh number thermal convection associated with mantle convection is chaotic, and we document the route to chaos. Deterministic chaos was first exhibited by the Lorenz equations, a three‐mode spatial Fourier expansion of the nonlinear thermal convection equations. This transition to chaos is characterized by the two nontrivial fixed points undergoing a reverse Hopf bifurcation into two unstable limit cycles. However, when the limit of infinite Prandtl number is taken for these equations, the fixed points are stable for all Rayleigh numbers. Is the stability of these fixed points a characteristic of infinite Prandtl number convection or is this an artifact of the truncation? To answer this question, we extend the expansion to 12 modes {n = 0, 1, 2; m = 1, 2, 3, 4} where n is the horizontal wave number and m is the vertical wave number to generate a set of 12 ordinary differential equations. We calculate the loci of roots of this system as a function of Rayleigh number, and we calculate several trajectories. At Rayleigh numbers between Ra = 0 and Ra = 4.140×104 we find pitchfork bifurcations to 14 separate nontrivial branches of solution, and we determine their stability by calculating the eigenvalues of the Jacobian matrix along each branch; we thus find 40 Hopf bifurcations. These fixed points and their bifurcations give excellent quantitative agreement with the trajectories, converging to stable fixed points at and below Ra = 4×104, and oscillating aperiodically at and above Ra = 4.5×104, where many of the Hopf bifurcations occur. Our conclusion is that thermal convection at infinite Prandtl number becomes chaotic by means of symmetry‐breaking pitchfork bifurcations and the appearance of the Hopf bifurcations which produce unstable periodic orbits.