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We study the effects of Prandtl number 
$\mathit {Pr}$ and Rayleigh number 
$\mathit {Ra}$ in two-dimensional Rayleigh–Bénard convection without boundaries, i.e. with periodic boundary conditions. For Prandtl numbers in the range 
$10^{-3} \leqslant \mathit {Pr} \leqslant 10^2$, the viscous dissipation scales as 
$\epsilon _\nu \propto \mathit {Pr}^{1/2}\mathit {Ra}^{-1/4}$, which is based on the observation that enstrophy 
$\langle {\omega ^2}\rangle \propto \mathit {Pr}^0 \mathit {Ra}^{1/4}$, and the Nusselt number tends to follow the ‘ultimate’ scaling 
$\mathit {Nu} \propto \mathit {Pr}^{1/2}\mathit {Ra}^{1/2}$ for all values of 
$\mathit {Pr}$ considered. The inverse cascade of kinetic energy forms the power-law spectrum 
$\hat {E}_u(k) \propto k^{-2.3}$, which is close to 
$k^{-11/5}$ proposed by the Bolgiano–Obukhov (BO) scaling. The potential energy flux is not constant, in contrast to one of the main assumptions underlying the BO phenomenology. So, the direct cascade of potential energy forms the power-law spectrum 
$\hat {E}_\theta (k) \propto k^{-1.2}$, which deviates from the expected 
$k^{-7/5}$. Finally, at 
$\mathit {Pr} \to 0$ and 
$\infty$, we find that the dynamics is dominated by vertically oriented elevator modes that grow without bound, even at high Rayleigh numbers and with large-scale dissipation present.
We call an 
$\alpha \in \mathbb {R}$ regainingly approximable if there exists a computable nondecreasing sequence 
$(a_n)_n$ of rational numbers converging to 
$\alpha $ with 
$\alpha - a_n < 2^{-n}$ for infinitely many 
${n \in \mathbb {N}}$. We also call a set 
$A\subseteq \mathbb {N}$ regainingly approximable if it is c.e. and the strongly left-computable number 
$2^{-A}$ is regainingly approximable. We show that the set of regainingly approximable sets is neither closed under union nor intersection and that every c.e. Turing degree contains such a set. Furthermore, the regainingly approximable numbers lie properly between the computable and the left-computable numbers and are not closed under addition. While regainingly approximable numbers are easily seen to be i.o. K-trivial, we construct such an 
$\alpha $ such that 
${K(\alpha \restriction n)>n}$ for infinitely many n. Similarly, there exist regainingly approximable sets whose initial segment complexity infinitely often reaches the maximum possible for c.e. sets. Finally, there is a uniform algorithm splitting regular real numbers into two regainingly approximable numbers that are still regular.
