The cooling box problem: convection with a quadratic equation of state

We investigate the convective cooling of a fluid with quadratic equation state by performing three-dimensional direct numerical simulations flow fixed top-boundary temperature, which is lower than initial temperature. consider temperatures near density maximum, where nonlinearity expected to be important. When nonlinear, resultant vertical transport heat fundamentally different and significantly predictions derived for linear state. Further, three dimensionless groups parameterize system: Rayleigh number ($\hbox{Ra}_0$), Prandtl (Pr), bottom water temperature $(T_B)$. further define an effective ($\hbox{Ra}_{eff} = Ra_0 \ T_B^2$), equivalent traditional used present predictive model flux, top boundary-layer thickness, turbulent kinetic energy system. show that this agrees well simulations. This could understand how quickly freshwater lakes cool in high latitude environments.