Linear and nonlinear dynamics in radiatively forced cold water

This paper investigates the linear and nonlinear dynamics of two-dimensional penetrative convection subjected to radiative volumetric thermal forcing, focusing on ice-covered freshwater systems. Linear stability analysis reveals how critical wavenumbers $k_c$ and Rayleigh numbers $Ra_c$ are influenced by the attenuation lengths and incoming heat flux. In this configuration, the system easily becomes unstable with a small $Ra_c$ , which is two decades smaller than that of the classical Rayleigh–Bénard convection problem, with typically $O(10)$ . Weakly nonlinear analysis figures out that this configuration is supercritical, contrasting with the subcritical case by Veronis ( Astrophys. J. , vol. 137, 1963, 641–663). Numerical bifurcation solutions are performed from the critical points and over several decades, up to $Ra \sim O(10^6)$ . This paper found that the system exhibits multiple steady solutions, and under certain specific conditions, a staircase temperature profile emerges. Meanwhile, we further discuss the influence of incoming heat flux and the Prandtl number $Pr$ on the primary bifurcation. Direct numerical simulations are also carried out, showing that heat is transported more efficiently via unsteady convection.