Linear and Weakly Nonlinear Stability of Combined Convection in a Rapidly Rotating Plane Layer in Planetary Convection Models

In the present study the combined convection in a rapidly rotating plane layer under the conditions that are characteristic of the near-polar regions in the planetary interiors is investigated. The combined thermal and compositional convection in a slowly rotating plane layer was previously considered for oceans, where convection is supported by thermal effects and is suppressed by compositional effects. The present work analyses the occurrence of convection by both of these effects with a predominant compositional effect in the Earth’s outer core and with various effects in the deep interiors of the known planets and moons. The self-consistent estimates of typical physical quantities give similarity coefficients for the small ratio dissipation/convection generation (s coincides with inverse Rayleigh number) and the ratio thermal convection/compositional convection (r). The third small coefficient (δ linked to the Ekman number) is the ratio of the characteristic size normal to the axis of rotation to the layer thickness. The effect of the important parameters δ and s on the stability of the combined thermal and compositional convection in a rapidly rotating plane layer is proposed in the literature by Starchenko (2017). To investigate the linear stability of this problem here, the normal mode method is employed. The critical values of  s and A (the critical wave number) observed to be depend on r for different values of δ and both Prandtl numbers that could imitate Solar System’s planets and moons at different ages. The obtained results coincide with those obtained by pioneers in the literature. The weakly nonlinear behaviour near to the primary instability threshold has been investigated using the spatiotemporal Landau-Ginzburg (LG) equation with cubic nonlinearity. Using the multiple scale analysis, the LG equation obtained and it is similar to those in the literature having different relaxation time, nonlinear coefficient, and coherence lengths. The heat transfer rate is studied using these coefficients. This equation is used to determine the domain for Eckhaus and zigzag as secondary instabilities.