Coherent heat transport in two-dimensional penetrative Rayleigh-Bénard convection

This paper investigates the steady coherent solutions, bifurcated from the linear stability of a stationary flow, in two-dimensional (2-D) penetrative convection. The results show that the thickness of the upper stably stratified layer, which is measured by θM (the dimensionless temperature at which the density is maximal, with 0 ≤ θM ≤ 1), plays an important role in the linear and nonlinear dynamics. First, we investigate steady solutions of fixed aspect ratio L = 2π/αc (where αc is the critical wavenumber). The results show that the instability is supercritical when θM < 0.4 and is subcritical when θM > 0.4. When θM > 0.4, the results show that the type of solution depends on the Prandtl number (Pr). For instance, when Pr ≲ 2.4 at θM = 0.5, the solution in one type of pair of convection cells does not exist, as the Rayleigh number Ra exceeds a critical value due to a saddle-node bifurcation. When Pr > 2.4, steady solutions can be found up to Ra = 108 for all θM, which exhibit the scaling of heat transfer (characterized by the Nusselt number Nu): Nu ∼ Ra1/4. Then, the optimal 2-D steady solutions are tracked up to Ra = 109 by varying the aspect ratio L, which shows that heat transfer roughly follows the Nu ∼ Raγ (γ ≈ 1/3) scaling in the regime of 107 < Ra < 109. It is interesting that the optimal temperature field has an arm-like horizontal structure when Pr < 10, while it has no significant horizontal structures when Pr > 10. Thus, the mean temperature in the mixing region is higher at large Pr. The steady solutions show that Nu ∼ Pr-1/12 for θM = 0 in a certain range of Pr by fixing the Rayleigh numbers, e.g. 1 < Pr < 10 for 2-D optimal steady solutions at Ra = 108 and 2 < Pr < 30 for 2-D steady solutions of fixed aspect ratio at Ra = 107. But when the Prandtl number is large or the upper stably stratified layer is thick, both the steady solutions of fixed aspect ratio and the 2-D optimal steady solutions are very weakly dependent on Pr.