Rapidly rotating thermal convection at low Prandtl number (Q2711499)

The author examines rapidly rotating convection at low Prandtl number \(\sigma\), where the onset of convection is oscillatory. Scaled equations are derived by considering distinguished limits, where \(\sigma^n\cdot\text{Ta}^{1/2}\sim 1\) but \(\sigma\to 0\) and \(\text{Ta}\to\infty\), for different values of \(n\geq 1\) (Ta is Taylor number). In the resulting asymptotic expansions in powers of \(\text{Ta}^{1/2}\) the leading-order equations, which are independent of \(n\), are solved to provide an analytic description of fully nonlinear convection. Three distinct asymptotic regimes are identified, distinguished by the relative importance of subdominant buoyancy and inertial terms. For the most interesting case, \(n=4\), the stability of different planforms near onset is investigated using a double expansion in powers of \(\text{Ta}^{-1/8}\) and in the amplitude of convection \(\varepsilon\). The case \(n=1\) is also analyzed.NEWLINENEWLINENEWLINEThe author shows that three contrasting asymptotic theories dominate different regions of \((\sigma,\text{Ta})\)-plane: the limit of rapid rotation only \((n\to \infty)\), and the cases \(n=4\) and \(n=1\), where the critical wavenumber at the onset of convection remains \(O(1)\). Very good agreement with numerical and experimental results from the available literature is found.