An extension of the 3D Lorenz model under the Gay-Lussac approximation

In this work, we explore a class of extensions to the 3D Lorenz (3DL) system by considering an alternative incompressible natural convection model. Famously, the 3DL system is recovered when the Oberbeck-Boussinesq (OB) approximation is applied to the 2D Rayleigh-Bénard problem. The OB model is incompressible, accounting for variations in fluid density exclusively in terms of buoyancy forces, which are modeled and closed by an equation of state that is linear in temperature. Gay-Lussac (GL) approximations relax OB by not discarding fluid density from the convection terms in the fluid momentum and heat equations. This class of models has been shown to resolve non-Boussinesq effects while preserving the incompressibility assumption, as well as the linear thermal expansion model. Herein, we present a class of dynamical systems that result from approaching the Rayleigh-Bénard problem with a GL model. We include linear stability analyses, Lyapunov spectra, and bifurcation diagrams.