Diffused-interface Rayleigh-Taylor instability with a nonlinear equation of state

Thermal convection in inland waters often occurs where the equation of state is highly nonlinear with temperature. We investigate the impact of this nonlinearity on the evolution of the Rayleigh-Taylor instability by analyzing the initial linear instability, the nonlinear plume growth, and the subsequent mixing resulting from this flow instability. The linear stability theory demonstrates that the thickness of the interface between the two layers of the Rayleigh-Taylor instability changes the wave number of maximum growth from the classical prediction. Our predicted wave number of maximum growth agrees well with two-dimensional direct numerical simulations of the diffused interface Rayleigh-Taylor instability. The nonlinear equation of state introduces asymmetry in the growing plumes about the density interface, preferentially generating kinetic energy in the lower layer. This asymmetry further introduces asymmetry in the location of the mixing. We analyze the energy evolution in the system and argue that the nonlinear equation of state will modify the distribution of heat in temperate lakes.