Blankenbach 3 revisited: intricate time-dependent patterns in a simple model of mantle convection

We evaluate time-dependent nature of mantle convection using a simple 2D Cartesian model with internal heating based on the benchmark case 3 in Blankenbach et al. (Geophysical Journal International, 1989). We are particularly interested in the bifurcation patterns of the Vrms-Nu phase plot for Rayleigh numbers (Ra) around the benchmark value (Ra = 2.16×105), but more information is disclosed when we go to higher Ra (up to 8×105). We also investigate the role of the boundary conditions, for which we change to periodic boundary conditions for a second bifurcation study. We find an intricate pattern in the behavior of the heat flow (as measured by the Nusselt number Nu(t)) and the kinetic energy (as measured by Vrms(t)) which include period doubling, break down of periodic into episodic flow and reorganization into periodic flow at higher Ra. Two patterns of bifurcation are found. One is the period doubling pattern, described in Blankenbach et al. 1989 and referred to as P2-P4 bifurcation. The period doubling results from the differentiation of existing limit points of the time series of Nu or Vrms. The other pattern is period-preserving, which is found at higher Ra number in this study. In the period-preserving bifurcation, the new limit points (peak and valley) of the Nu and Vrms time series showed up as a twist in the monotonic intervals between a peak and valley. In this case the period doesn’t change. Both of the two patterns are observed in the models with the two types of boundary conditions (reflective and periodic). At a given Ra, different solutions can be obtained with different initial conditions. The initial condition is usually a solution with its Ra in the neighborhood, and with this neighborhood searching method, we were able to span the bifurcation plot (Ra-limit points of Nu(t) or Vrms(t)) to the range of Ra = 1×105~8×105 with both two boundary conditions. In this process, hysteresis is observed as expected in dynamic system, and the overlap of different trend of solutions are large enough to demonstrate tough competition between them. Limit points of Vrms time series obtained at same conditions with Blanckenbach case 3 with different Rayleigh numbers, showing bifurcation.