Based on the fully compressible Navier-Stokes equations, the linear stability of thermal convection in rapidly rotating spherical shells of various radius ratios eta is studied for a wide range of Taylor number Ta, Prandtl number Pr and the number of density scale height N-rho. Besides the classical inertial mode and columnar mode, which are widely studied by the Boussinesq approximation and anelastic approximation, the quasi-geostrophic compressible mode is also identified in a wide range of N-rho and Pr for all eta considered, and this mode mainly occurs in the convection with relatively small Pr and large N.. The instability processes are classified into five categories. In general, for the specified wavenumber m, the parameter space (Pr, N-rho) of the fifth category, in which the base state loses stability via the quasi-geostrophic compressible mode and remains unstable, shrinks as eta increases. The asymptotic scaling behaviours of the critical Rayleigh numbers Ra-c and corresponding wavenumbers m(c) to Ta are found at different eta for the same instability mode. As eta increases, the flow stability is strengthened. Furthermore, the linearized perturbation equations and Reynolds-Orr equation are employed to quantitatively analyse the mechanical mechanisms and flow instability mechanisms of different modes. In the quasi-geostrophic compressible mode, the time-derivative term of disturbance density in the continuity equation and the diffusion term of disturbance temperature in the energy equation are found to be critical, while in the columnar and inertial modes, they can generally be ignored. Because the time-derivative term of the disturbance density in the continuity equation cannot be ignored, the anelastic approximation fails to capture the instability mode in the small-Pr and large-N-rho system, where convection onset is dominated by the quasi-geostrophic compressible mode. However, all the modes are primarily governed by the balance between the Coriolis force and the pressure gradient, based on the momentum equation. Physically, the most important difference between the quasi-geostrophic compressible mode and the columnar mode is the role played by the disturbance pressure. The disturbance pressure performs negative work for the former mode, which appears to stabilize the flow, while it destabilizes the flow for the latter mode. As eta increases, in the former mode the relative work performed by the disturbance pressure increases and in the latter mode decreases.

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier-Stokes equations. Compressibility is quantified by the number of density scale heights N-rho, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index n = 1.499 < na = 1.5 is considered, where na is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number Pr and to N-rho. For large Pr and small N-rho, the quasi-geostrophic columnar mode loses stability first; while for relatively small Pr a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small Pr and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers Ra for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical Ra for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative Ra. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-Ra and large-N-rho regime.

The influences of non-Oberbeck-Boussinesq (NOB) effects on flow instabilities and bifurcation characteristics of Rayleigh-Benard convection are examined. The working fluid is air with reference Prandtl number Pr = 0.71 and contained in two-dimensional rigid cavities of finite aspect ratios. The fluid flow is governed by the low-Mach-number equations, accounting for the NOB effects due to large temperature difference involving flow compressibility and variations of fluid viscosity and thermal conductivity with temperature. The intensity of NOB effects is measured by the dimensionless temperature differential epsilon. Linear stability analysis of the thermal conduction state is performed. An epsilon(2) scaling of the leading-order corrections of critical Rayleigh number Ra-cr and disturbance growth rate sigma due to NOB effects is identified, which is a consequence of an intrinsic symmetry of the system. The influences of weak NOB effects on flow instabilities are further studied by perturbation expansion of linear stability equations with regard to epsilon, and then the influence of aspect ratio A is investigated in detail. NOB effects are found to enhance (weaken) flow stability in large (narrow) cavities. Detailed contributions of compressibility, viscosity and buoyancy actions on disturbance kinetic energy growth are identified quantitatively by energy analysis. Besides, a weakly nonlinear theory is developed based on centre-manifold reduction to investigate the NOB influences on bifurcation characteristics near convection onset, and amplitude equations are constructed for both codimension-one and -two cases. Rich bifurcation regimes are observed based on amplitude equations and also confirmed by direct numerical simulation. Weakly nonlinear analysis is useful for organizing and understanding these simulation results.