Thermal convection in rapidly rotating, self-gravitating Boussinesq fluid spheres is characterized by three parameters: the Rayleigh number $R$, the Prandtl number Pr and the Ekman number $E$. Two different asymptotic limits were considered in the previous studies of the linear problem. In the double limit $E \,{\ll}\, 1$ and $\hbox{\it Pr}/E \,{\gg}\, 1$, the local asymptotic theory showed that the convective motion is strongly non-axisymmetric, columnar, highly localized and described by the asymptotic scalings, $({1}/{s}) {\zpt}/{\zpt \phi} \,{=} {O}\big(E^{-1/3}\big),\; {\zpt}/{\zpt z} \,{=}\, {O}(1), \; R_c\,{=}\,{O}\big(E^{-1/3}\big)$, where $R_c$ denotes the critical Rayleigh number and $(s,\phi,z)$ are cylindrical polar coordinates with the axis of rotation at $s\,{=}\,0$. A global asymptotic theory with novel features for the limit $E \,{\ll}\, 1$ and $\hbox{\it Pr}/E \,{\gg}\, 1$, indicating the radial asymptotic scaling ${\zpt}/{\zpt s} \,{=}\, {O}\big(E^{-1/3}\big)$, was recently developed by Jones et al. (J. Fluid Mech. vol. 405, 2000, p. 157). In the different double limit $E \,{\ll}\, 1$ and $ \hbox{\it Pr}/E \,{\ll}\, 1 $, an asymptotic theory for the onset of convection building upon the theory of inertial waves was developed by Zhang (J. Fluid Mech. vol. 268, 1994 p. 211). It was shown that the convective motion at the leading-order approximation is represented by a single inertial-wave mode with a quadratic polynomial of $s$ and $z$, obeying the asymptotic dependence ${\zpt}/{\zpt s} \,{\sim}\, ({1}/{ s}) {\zpt}/{\zpt \phi} \,{=}\, {O}(1), {\zpt}/{\zpt z} \,{=}\, {O}(1)$ and $ R_c\,{=}\,{O}(E)$ for stress-free spheres.
There exist no simple asymptotic scalings for $ E \,{\ll}\, 1 $ appropriate to all values of $\hbox{\it Pr}/E$. For an arbitrary small but non-zero $E$, the highly localized convection spreads out spatially with decreasing $\hbox{\it Pr}$, suggesting that the scaling laws such as ${\zpt}/{\zpt s} \,{=}\, {O}\big(E^{-1/3}\big)$ are no longer valid when $\hbox{\it Pr}/E$ is not sufficiently large. This paper represents an attempt to develop a new asymptotic method for the analysis of convection in rapidly rotating spheres valid for asymptotically small $E$ and for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$. The new method is based on the following three hypotheses. The first is that the leading-order velocity of convection for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$ at $E \,{\ll}\, 1$ is represented by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes convectively excited and sustained. Secondly, we assume that the convective motion for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$ at $E \,{\ll}\, 1$ always has columnar structure, i.e. ${\zpt}/{\zpt z} \,{\sim}\, {O}(1)$, but without the general asymptotic scalings in the radial and azimuthal direction. Thirdly, we assume that there always exists a boundary flow that is non-zero only in the Ekman boundary layer on the bounding spherical surface and plays an important role even in the case of stress-free boundaries. Comparison between the result of the new method and the corresponding fully numerical simulation demonstrates a satisfactory quantitative agreement for all values of $0 \,{\le}\,\hbox{\it Pr}/E \,{\le}\,{O}(10^6)$ when $ {O} (10^{-5}) \le E \,{\le}\,{O}(10^{-6})$. The new method is asymptotic in the sense that it is valid only for an asymptotically small $E \,{\ll}\, 1$.
In addition to the linear problem of thermal convection in rapidly rotating spheres, the corresponding weakly nonlinear problem is also solved to obtain an analytical expression for the convection-driven differential rotation generated by the nonlinear interaction of quasi-geostrophic-inertial-wave modes through the Reynolds stresses. The new method not only reveals the underlying nature of thermal convection in rapidly rotating spheres but also unites the two previously disjointed subjects in rotating fluids: the inertial-wave problem and the convective instability problem.