The impact of a nonlinear equation of state on the convective instability of a radiatively heated system

Journal article

We consider convection in an internally heated system with a nonlinear equation of state using linear stability analysis, Direct Numerical Simulation (DNS) and weakly nonlinear analysis. Motivated by melt ponds on Arctic sea ice, we consider a fluid layer subject to heating from radiative absorption, an isothermal lower boundary, and an imperfectly conducting top boundary. A modified Fourier spectral method is employed in the linear stability analysis to ensure numerical convergence. We perform two-dimensional DNS for a wide range of parameters, alongside representative three-dimensional cases. Both linear theory and the DNS show that the nonlinear equation of state significantly influences the instability characteristics. We define a dimensionless parameter 𝜅𝑛 to quantify the level of nonlinearity of the equation of state, so that the Rayleigh number 𝑅𝑎 and 𝜅𝑛𝑃𝑟𝑅𝑎2 act as a measure of the buoyancy-driven forcing caused by the linear and the nonlinear parts of the equation of state, respectively. Here 𝑃𝑟 is the Prandtl number. With sufficiently large 𝑅𝑎, and increasing |𝜅𝑛|, the system undergoes a transition between different modal structures. There is bottom-up convection from the lower boundary with small |𝜅𝑛|, and shallow top-down convection from the top boundary with large |𝜅𝑛|. For intermediate |𝜅𝑛| and increasing 𝑅𝑎 we find successive alternating windows of stability, instability, reemergence of stability, and finally instability. A weakly nonlinear analysis indicates that the bifurcations are supercritical. The implications for mixing in Arctic melt ponds are discussed. The net heat flux across the atmosphere-pond interface varies depending on the flow regime.

Authors: Zhang, Y; Wells, A

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Journal of Fluid Mechanics
• Acceptance date: 2026-05-26
• EISSN: 1469-7645
• ISSN: 0022-1120
• Language: English