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

Abstract

Abstract Content of image described in text. We consider convection in an internally heated system with a nonlinear equation of state (EOS) 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 EOS significantly influences the instability characteristics. We define a dimensionless parameter kappa Subscript n κn $\kappa _n$ to quantify the level of nonlinearity of the EOS, so that the Rayleigh number italic Ra Ra $ \textit{Ra}$ and kappa Subscript n Baseline upper P r upper R a squared κnPrRa2 $\kappa _n PrRa^2$ (where italic probability Pr $ \textit{Pr}$ is the Prandtl number) act as a measure of the buoyancy-driven forcing caused by the linear and nonlinear parts of the EOS, respectively. With sufficiently large italic Ra Ra $ \textit{Ra}$ , and increasing StartAbsoluteValue kappa Subscript n Baseline EndAbsoluteValue |κn| $|\kappa _n|$ , the system undergoes a transition between different modal structures. There is bottom-up convection from the lower boundary with small StartAbsoluteValue kappa Subscript n Baseline EndAbsoluteValue |κn| $|\kappa _n|$ , and shallow top-down convection from the top boundary with large StartAbsoluteValue kappa Subscript n Baseline EndAbsoluteValue |κn| $|\kappa _n|$ . For intermediate StartAbsoluteValue kappa Subscript n Baseline EndAbsoluteValue |κn| $|\kappa _n|$ and increasing italic Ra Ra $ \textit{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.