Toroidal and poloidal energy in rotating Rayleigh–Bénard convection

We consider rotating Rayleigh–Bénard convection of a fluid with a Prandtl number of $\mathit{Pr}=0.8$ in a cylindrical cell with an aspect ratio ${\it\Gamma}=1/2$. Direct numerical simulations (DNS) were performed for the Rayleigh number range $10^{5}\leqslant \mathit{Ra}\leqslant 10^{9}$ and the inverse Rossby number range $0\leqslant 1/\mathit{Ro}\leqslant 20$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy-dominated regime occurring while the toroidal energy $e_{tor}$ is not affected by rotation and remains equal to that in the non-rotating case, $e_{tor}^{0}$. Second, a rotation-influenced regime, starting at rotation rates where $e_{tor}>e_{tor}^{0}$ and ending at a critical inverse Rossby number $1/\mathit{Ro}_{cr}$ that is determined by the balance of the toroidal and poloidal energy, $e_{tor}=e_{pol}$. Third, a rotation-dominated regime, where the toroidal energy $e_{tor}$ is larger than both $e_{pol}$ and $e_{tor}^{0}$. Fourth, a geostrophic regime for high rotation rates where the toroidal energy drops below the value for non-rotating convection.