2. Numerical set-up and analysis methods

We perform DNSs of RBC in a cylindrical sample rotating around its vertical axis. The governing equations in RBC encompass the balances of mass, momentum and energy conservation. The influence of constant rotation around the axis is included by considering the Coriolis force in the momentum equations. For an incompressible fluid the governing equations under the Oberbeck–Boussinesq approximation are solved in their dimensionless form (e.g. Zhong et al. Reference Zhong, Stevens, Clercx, Verzicco, Lohse and Ahlers2009)

(2.1a–c)\begin{equation} \left.\begin{gathered} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0\\ \frac{\mathrm{d}\boldsymbol{u}}{\mathrm{d}t}=-\boldsymbol{\nabla} P + \sqrt{\frac{\textit{Pr}}{ {\textit{Ra}}}}\nabla^2\boldsymbol{u}+\varTheta\boldsymbol{e}_z-\frac{1}{ {\textit{Ro}}}\boldsymbol{e}_z \times\boldsymbol{u}\\ \frac{\mathrm{d}\varTheta}{\mathrm{d}t}=\frac{1}{\sqrt{\textit{Pr}\, {\textit{Ra}}}}\nabla^2\varTheta. \end{gathered}\right\} \end{equation}

Here, $\boldsymbol {u}$, $P$ and $\varTheta$ are the normalized velocity, pressure and temperature fields, respectively. The equations are normalized by the height of the cylinder $H$ and the free-fall velocity $U_0=\sqrt {\alpha g \Delta T H}$, where $\alpha$ is the isobaric thermal expansion coefficient, $g$ the gravitational acceleration and $\Delta T$ the temperature difference between upper and lower plate. The temperature is normalized as $\varTheta =({T-T_{{top}}})/{\Delta T}\in [0,1]$. The pressure field P is reduced by the hydrostatic balance and centrifugal contributions. The control parameters in the equations then are the Prandtl number $\textit {Pr}$, the Rayleigh number $ {\textit {Ra}}$ and the inverse Rossby number $ {\textit {Ro}}^{-1}$

(2.2a–c)\begin{equation} \textit{Pr}=\nu/\kappa;\quad {\textit{Ra}}=\alpha g \Delta T H^{3}/(\nu\kappa) ;\quad {\textit{Ro}}^{{-}1}=2\varOmega H/U_0. \end{equation}

Here, $\nu$ is the kinematic viscosity, $\kappa$ the thermal diffusivity and $\varOmega$ the rotation rate. The inverse Rossby number $ {\textit {Ro}}^{-1}$ can alternatively be expressed in terms of the Ekman number $ {\textit {Ek}}$: $ {\textit {Ro}}^{-1}=2\sqrt {\textit {Pr}/ {\textit {Ra}}}\, {\textit {Ek}}^{-1}$. Finally, the confinement parameter $\varGamma ^{-1}$ defines the cylinder size of height $H$ and diameter $D$:

(2.3)\begin{equation} \varGamma^{{-}1}=H/D. \end{equation}

No-slip boundary conditions are imposed on the isothermal top and bottom plates and the adiabatic sidewall. In all simulations the Prandtl number is fixed at $\textit {Pr}=4.38$ ($\approx$ water), the Rayleigh number is chosen in the range $2\times 10^{8}\leqslant {\textit {Ra}}\leqslant 7\times 10^{9}$. Among all $ {\textit {Ra}}$ we cover a parameter space of $0\leqslant {\textit {Ro}}^{-1}\leqslant 40$ and $2\leqslant \varGamma ^{-1}\leqslant 32$.

The governing equations (2.1) are solved by a central second-order accurate finite-difference discretization on a staggered grid as presented in Verzicco & Orlandi (Reference Verzicco and Orlandi1996) and Verzicco & Camussi (Reference Verzicco and Camussi1997, Reference Verzicco and Camussi1999). The code has been often validated (e.g. Kooij et al. Reference Kooij, Botchev, Frederix, Geurts, Horn, Lohse, van der Poel, Shishkina, Stevens and Verzicco2018). For a sufficient resolution of the Kolmogorov scales in the entire domain, we increase the number of grid points in the vertical direction from $N_z=256$ for $ {\textit {Ra}}=2\times 10^{8}$ up to $N_z=768$ for $ {\textit {Ra}}=7\times 10^{9}$. Additionally, we use a stretched grid in the vertical and radial directions to ensure that the resolution of the boundary layers at the plates and the sidewall fulfil the required criteria given in Shishkina et al. (Reference Shishkina, Stevens, Grossmann and Lohse2010). The dynamic time stepping in our simulations is controlled by a maximum Courant number and a maximum time step. The numerical parameters per simulation set are summarized in Appendix A.

The global key response parameter of the system is the dimensionless heat transport given by the Nusselt number $ {\textit {Nu}}=Q H/(\kappa \Delta T)$ with the heat flux $Q$ from the bottom to the top plate. We compute $ {\textit {Nu}}\equiv \left \langle {\textit {Nu}}(t)\right \rangle _t$ directly from the vertical gradient of the dimensionless temperature $\varTheta$ as the average $\left \langle \cdot \right \rangle$ over both plates

(2.4)\begin{equation} {\textit{Nu}}(t)=\left\langle-\partial_z\left\langle\varTheta(t)\right\rangle_{r,\vartheta}\right\rangle_{z=\{0,1\}}. \end{equation}

For comparison between different $ {\textit {Ra}}$, we present all heat transport data normalized as $ {\textit {Nu}}/ {\textit {Nu}}_0$, where $ {\textit {Nu}}_0( {\textit {Ra}}, {\textit {Ro}}^{-1}=0,\varGamma ^{-1}=1)$ serves as $ {\textit {Ra}}$-dependent reference value of the (mostly) non-stabilized system, i.e. a non-rotating, sufficiently wide cylinder. The reference values $ {\textit {Nu}}_0$ are known from various experiments and numerical simulations, and are well described by the Grossmann–Lohse theory (Grossmann & Lohse Reference Grossmann and Lohse2000, Reference Grossmann and Lohse2001; Stevens et al. Reference Stevens, van der Poel, Grossmann and Lohse2013b). Further, we use the root mean square (r.m.s.) of the normalized temporal Nusselt number fluctuations to quantify the temporal stability of the flow

(2.5)\begin{equation} {\textit{Nu}}_{rms}=\sqrt{\left\langle\left(\frac{{\textit{Nu}}(t)}{\left\langle{\textit{Nu}}\right\rangle_t}-1\right)^{2}\right\rangle_t}. \end{equation}

To characterize the flow we use the directional Reynolds numbers $\textit {Re}_{r,\vartheta,z}\equiv \left \langle \textit {Re}_{r,\vartheta,z}(t)\right \rangle _t$ based on the dimensionless r.m.s. velocities $\sqrt {\left \langle u_{i}^{2}\right \rangle _{r,\vartheta,z}}$

(2.6)\begin{align} \textit{Re}_{i}(t)&=\frac{U_0 H}{\nu}\sqrt{\left\langle u_{i}^{2}(t)\right\rangle_{r,\vartheta,z}}:\ i=\{r,\vartheta,z\} \nonumber\\ &=\sqrt{\frac{{\textit{Ra}}}{\textit{Pr}}}\sqrt{\left\langle u_{i}^{2}(t)\right\rangle_{r,\vartheta,z}}.\end{align}

These Reynolds numbers allow the distinction of the strength of fluid motion in the different directions, i.e. radial, azimuthal and vertical directions. Consequently, their ratios can depict the predominant flow motion and will help to classify characteristic flow states. Temporal averaging starts when the statistically stationary state has been reached.

The ratio of thermal to kinetic boundary layer thicknesses plays a crucial role in Ekman pumping related heat transport enhancement (Stevens et al. Reference Stevens, Clercx and Lohse2010b; Yang et al. Reference Yang, Verzicco, Lohse and Stevens2020). We determine the thermal boundary layer thickness $\lambda _{\varTheta }$ from the vertical profile of the horizontally averaged temperature (e.g. Stevens et al. Reference Stevens, Clercx and Lohse2010a), where $\lambda _{\varTheta }$ is defined as the intersection between the temperature gradient at the plate with a linear fit to the temperature profile in the bulk ($0.2\leqslant z\leqslant 0.8$). Following Stevens et al. (Reference Stevens, Clercx and Lohse2010a) and Yang et al. (Reference Yang, Verzicco, Lohse and Stevens2020), the kinetic boundary layer thickness $\lambda _u$ is estimated as twice the height of the peak location of the horizontally averaged quantity $\left \langle \boldsymbol {u}\boldsymbol {\cdot }\nabla ^{2}\boldsymbol {u}\right \rangle _H$. The horizontal averages involve only 90 % of the radial distance from the axis, which in our case worked best to consistently exclude effects of the sidewall boundary layer (Wagner, Shishkina & Wagner Reference Wagner, Shishkina and Wagner2012). The reported values are averaged over top and bottom boundary layers.

3. Heat transport maxima under rotation and confinement

First, we keep $ {\textit {Ra}}=7\times 10^{8}$ (and $\textit {Pr}=4.38$) fixed and vary the confinement parameter $\varGamma ^{-1}$ and the rotation rate $ {\textit {Ro}}^{-1}$. When, on the one hand, $\varGamma ^{-1}$ is varied in the absence of rotation $ {\textit {Ro}}^{-1}=0$, the normalized heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$ is largest for $\varGamma ^{-1}=12$ (figure 1a). When, on the other hand, $ {\textit {Ro}}^{-1}$ is varied in a relatively wide cylinder with $\varGamma ^{-1}=3$, the largest $ {\textit {Nu}}/ {\textit {Nu}}_0$ is reached at $ {\textit {Ro}}^{-1}\approx 8$. Both are known as the heat transport enhancement by either moderate confinement (e.g. Huang et al. Reference Huang, Kaczorowski, Ni and Xia2013; Chong et al. Reference Chong, Huang, Kaczorowski and Xia2015) or moderate rotation (e.g. Zhong et al. Reference Zhong, Stevens, Clercx, Verzicco, Lohse and Ahlers2009; Kunnen et al. Reference Kunnen, Stevens, Overkamp, Sun, van Heijst and Clercx2011). However, surprisingly, the two-dimensional parameter subspace of $\varGamma ^{-1}$ and $ {\textit {Ro}}^{-1}$ reveals a strongly non-symmetric enhancement of the heat transport when both confinement and rotation are present simultaneously (figure 1a). Hence, the enhancing effects of both types of stabilization do not simply superpose with each other. Instead they interact in a more complex way.

Figure 1. Heat transport maxima and flow characteristics in the parameter space of rotation $ {\textit {Ro}}^{-1}$ and confinement $\varGamma ^{-1}$ for $ {\textit {Ra}}=7\times 10^{8}$. A, B and C mark the positions of the confinement (§ 3.1), double-vortex (§ 3.2) and single-vortex maxima (§ 3.3), respectively. Grey, green and red lines show the transitions between the (sub-)regimes of prominent flow characteristics. (a) Normalized heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$ (circles, data points; background, cubic interpolation). (b) Regimes of prominent flow motion based on linear interpolation of $\textit {Re}_{\vartheta,r,z}$ data. The grey area depicts the buoyancy-dominated regime ($\textit {Re}_z>\textit {Re}_H=(\textit {Re}_{\vartheta }^{2}+\textit {Re}_{r}^{2})^{1/2}$). The coloured areas belong to the rotation-controlled regime ($\textit {Re}_H>\textit {Re}_z$). In the green sub-regime the characteristic flow motion ($\textit {Re}_{\vartheta }>\textit {Re}_r>\textit {Re}_z$) indicates double-vortex flow. In the red sub-regime suppressed radial motion (see (e)) indicates single-vortex flow. The symbol colour shows again the heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$. (c) Mean vertical temperature gradient in the bulk $\langle \partial _z\langle \varTheta \rangle _{r,\vartheta,t}\rangle _{0.2\leqslant z\leqslant 0.8}$ (circles, data points; background, linear interpolation). (d) Temporal stability of the flow based on $ {\textit {Nu}}_{rms}$ ((2.5), circles, data points; background, linear interpolation). (e) Ratio of azimuthal to radial Reynolds numbers $\textit {Re}_{\vartheta }/\textit {Re}_r$ (circles, data points; background, linear interpolation).

Our simulations reveal three separate maxima of the heat transport (figure 1a). Maximum A is the heat transport enhancement by moderate confinement in the absence of rotation, which vanishes already under weak rotation. Therefore, we will hereafter refer to it as the confinement maximum. On the contrary, the enhancement by moderate rotation $ {\textit {Ro}}^{-1}$ in large domains does not directly vanish with increasing confinement $\varGamma ^{-1}$ (figure 1a). It further extends into the $( {\textit {Ro}}^{-1},\varGamma ^{-1})$ parameter space and even increases up to maximum B, which is the largest heat transport achieved for $ {\textit {Ra}}=7\times 10^{8}$. Thereby, the range of $ {\textit {Ro}}^{-1}$, which enhances the heat transport, shifts towards faster rotation with increasing $\varGamma ^{-1}$, and separates from the confinement maximum. The onset of heat transport enhancement strongly correlates with $\textit {Re}_z=\textit {Re}_H=(\textit {Re}_{\vartheta }^{2}+\textit {Re}_{r}^{2})^{1/2}$, the transition from buoyancy-dominated to rotation-controlled regime (figure 1b; thick grey line), which will be further discussed in § 4. Even more surprisingly, we observe a third maximum C at the tip of the enhancement region around maximum B.

All maxima are associated with an individual (sub-)regime of the fluid motion (figure 1b), which characterizes the flow pattern around each maximum. At the confinement maximum the flow forms vertically coherent buoyant plumes (figure 2a). At maximum B the flow is organized in two stable vortices (figure 3a). Hence, we will hereafter refer to it as the double-vortex maximum. At maximum C the flow is characterized by one stable central vortex (figure 4a). Accordingly, we will refer to it as the single-vortex maximum. Next, we present each of the three maxima and its specific flow dynamics separately in detail (§§ 3.1–3.3).

Figure 2. Flow characteristics at the confinement maximum at $ {\textit {Ra}}=7\times 10^{8}$ ($\varGamma ^{-1}=12$, $ {\textit {Ro}}^{-1}=0$): (a) snapshot of the temperature field, (b) temporal evolution of $ {\textit {Nu}}(t)$ at the top and bottom plate, (c) temporal evolution of $\textit {Re}(t)$ for each velocity component.

Figure 3. Flow characteristics at the steady double-vortex maximum at $ {\textit {Ra}}=7\times 10^{8}$ ($\varGamma ^{-1}=5, {\textit {Ro}}^{-1}=12.5$): (a) snapshot of the temperature field, (b) temporal evolution of $ {\textit {Nu}}(t)$ at the top and bottom plate, (c) temporal evolution of $\textit {Re}(t)$ for each velocity component.

Figure 4. Flow characteristics at the steady single-vortex maximum at $ {\textit {Ra}}=7\times 10^{8}$ ($\varGamma ^{-1}=8, {\textit {Ro}}^{-1}=20$): (a) snapshot of the temperature field, (b) temporal evolution of $ {\textit {Nu}}(t)$ at the top and bottom plate, (c) temporal evolution of $\textit {Re}(t)$ for each velocity component.

4. Interfering effects for heat transport enhancement

The central mechanism for heat transport enhancement under rotation is Ekman pumping (e.g. Rossby Reference Rossby1969). The common view is that the onset of Ekman pumping causes the transition from a large-scale circulation to vertically aligned vortices, which can result in heat transport enhancement (Julien et al. Reference Julien, Legg, McWilliams and Werne1996). Although it has been recently observed that the onset of heat transport enhancement and the change in flow dynamics have slightly different onsets (Alards et al. Reference Alards, Kunnen, Stevens, Lohse, Toschi and Clercx2019), both are certainly closely related to each other.

In this work, we define the transition from buoyancy-dominated to rotation-controlled regime by the change of the predominant fluid motion from vertical to horizontal, i.e. at $\textit {Re}_{z}=\textit {Re}_{H}$ (figure 1b; thick grey line). The transition ${\textit {Re}_{z}=\textit {Re}_{H}=(\textit {Re}_{\vartheta }^{2}+\textit {Re}_{r}^{2})^{1/2}}$ describes the onset of heat transport enhancement in very confined systems significantly better than the commonly used transition of the kinetic boundary layer from Prandtl–Blasius type to Ekman type (Appendix B: figure 12). We note that, in this context, any Ekman pumping related increase of $ {\textit {Nu}}$ with $ {\textit {Ro}}^{-1}$ is seen as heat transport enhancement independent of whether $ {\textit {Nu}}/ {\textit {Nu}}_0>1$ or not. In our view, it is reasonable that vertical motion is predominant in the buoyancy-dominated regime, and horizontal motion takes over in the rotation-controlled regime, when vortical structures dominate the flow. The vortical structures extend the vertical transport induced by Ekman pumping through the bulk, which thereby increases the heat transport. We observe that all simulations with a rotation enhanced heat transport are characterized by $\textit {Re}_{H}>\textit {Re}_{z}$, and that the onset is located just beyond this transition (figure 1b).

Both the transition $\textit {Re}_{z}=\textit {Re}_{H}$ and the onset of heat transport enhancement simultaneously shift towards faster rotation $ {\textit {Ro}}^{-1}$ with increasing confinement $\varGamma ^{-1}$. With increasing $\varGamma ^{-1}$, the cylinder diameter approaches the critical length scale of convective instability $L_c=4.82\times Ek^{1/3}$ for rotating RBC (Chandrasekhar Reference Chandrasekhar1961; Kunnen et al. Reference Kunnen, Ostilla-Mónico, van der Poel, Verzicco and Lohse2016). We assume that the formation of vortical flow structures requires a minimal lateral size, i.e. a maximal confinement $\varGamma ^{-1}_{max}(L_c)$, which can be reached with faster rotation. Consequently, the transitional rotation rate $ {\textit {Ro}}^{-1}_{BD\text {-}RC}$ for the onset of heat transport enhancement is strongly $\varGamma ^{-1}$-dependent in slender cylinders, but might be insensitive to $\varGamma ^{-1}$ in sufficiently wide domains. Several approaches employ a Ginzburg–Landau model to account for the $\varGamma ^{-1}$-dependence (e.g. Weiss et al. Reference Weiss, Stevens, Zhong, Clercx, Lohse and Ahlers2010; Weiss & Ahlers Reference Weiss and Ahlers2011b), but ignore any $ {\textit {Ra}}$-dependence, which will be addressed in § 5.

Ideally, an equal thickness of thermal and kinetic boundary layer is supposed to maximize the heat transport in rotating RBC (e.g. Julien et al. Reference Julien, Legg, McWilliams and Werne1996; King et al. Reference King, Stellmach, Noir, Hansen and Aurnou2009), since Ekman pumping becomes most efficient in ejecting heat from the boundary layers into the columnar vortices (Stevens et al. Reference Stevens, Clercx and Lohse2010b). However, this is only valid up to a certain $ {\textit {Ra}}$ and confinement (Yang et al. Reference Yang, Verzicco, Lohse and Stevens2020). Similar to Yang et al. (Reference Yang, Verzicco, Lohse and Stevens2020), we test this assumption by mapping the heat transport onto the ratio of thermal and kinetic boundary layer thicknesses $\lambda _{\varTheta }/\lambda _u$ for fixed confinement $\varGamma ^{-1}$ (figure 5). Thereby, an increasing $\lambda _{\varTheta }/\lambda _u$ is generally related to an increasing $ {\textit {Ro}}^{-1}$. Accordingly, our widest cylinder ($\varGamma ^{-1}=3$) shows a relatively symmetric heat transport enhancement around $\lambda _{\varTheta }/\lambda _u=1$ (figure 5a). With increasing confinement ($\varGamma ^{-1}=4$), the maximal heat transport at $\lambda _{\varTheta }/\lambda _u\approx 1$ significantly increases but also gains some asymmetry around the enhancement peak (figure 5a). Thereby, the large heat transport coincides with the presence of double-vortex flow (figure 5d; green symbols). For confinement $\varGamma ^{-1}_{2VM}=5$, at which the double-vortex maximum is obtained, the peak is still located close to $\lambda _{\varTheta }/\lambda _u\!=\!1$, but with a rather sharp onset just at $\lambda _{\varTheta }/\lambda _u\approx 1$ (figure 5b). The largest heat transport is again associated with stable double-vortex flow (figure 5e; green symbols). In slender cylinders ($\varGamma ^{-1}>\varGamma ^{-1}_{2VM}$), no enhancement can be observed around $\lambda _{\varTheta }/\lambda _u=1$ (figure 5c). Still, large heat transport is observed twice: (i) rotation induced enhancement only appears at $\lambda _{\varTheta }/\lambda _u>1$ again with an extremely sharp onset. The largest heat transport again coincides with double-vortex or single-vortex flow (figure 5f; green and red symbols). (ii) Large heat transport at $\lambda _{\varTheta }/\lambda _u\approx 0.6<1$ corresponds to the non-rotating cases and the enhancement due to confinement (§ 3.1), which is different from the expectation $\lambda _{\varTheta }/\lambda _u\approx 1$ based on the unifying view of Chong et al. (Reference Chong, Yang, Huang, Zhong, Stevens, Verzicco, Lohse and Xia2017). We explain this deviation by the different definitions used to determine the kinetic boundary layer thickness. This, however, seems to affect mainly the estimates for the non-rotating confined cases. Nonetheless, the kinetic and thermal boundary layers themselves evolve in general as expected (see Appendix B: figures 12b, 13).

Figure 5. Normalized heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$ mapped onto the ratio of thermal and kinetic boundary layers $\lambda _\varTheta /\lambda _u$ for various $\varGamma ^{-1}$ at $ {\textit {Ra}}=7\times 10^{8}$: (b) for $\varGamma ^{-1}_{2VM}=5$, where the double-vortex maximum is observed, (a) for less confinement $\varGamma ^{-1}<\varGamma ^{-1}_{2VM}$, i.e. wider cylinders than in (b), (c) for more confinement $\varGamma ^{-1}>\varGamma ^{-1}_{2VM}$, i.e. more slender cylinders than in (b). (d–f) Same data as in (a–c), respectively, but symbols coloured in their corresponding regime of prominent flow motion (figure 1b): grey – buoyancy-dominated regime, blue – rotation-controlled regime, green – rotation-controlled with double-vortex flow (figure 3a), red – rotation-controlled with single-vortex flow (figure 4a). The vertical grey line marks the most beneficial boundary layer ratio $\lambda _\varTheta /\lambda _u=1$. The dashed grey line (no specific scaling) serves as guide for the eye. The coloured and grey lines connecting the data points in (a–c) and (d–f), respectively, follow increasing rotation $ {\textit {Ro}}^{-1}$ per confinement $\varGamma ^{-1}$.

The sharp onsets for $\varGamma ^{-1}\geqslant \varGamma ^{-1}_{2VM}$ result from a shifted transition from the buoyancy-dominated to the rotation-controlled regime. In less confined domains (${\varGamma ^{-1}<\varGamma ^{-1}_{2VM}}$), the rotation-controlled regime already begins at $\lambda _{\varTheta }/\lambda _u\ll 1$ and thus the efficiency of Ekman pumping can symmetrically increase and decrease around $\lambda _{\varTheta }/\lambda _u=1$ (figure 5d; blue symbols). However, as long as the flow is buoyancy-dominated, no vortical structure can profit from Ekman pumping to enhance the heat transport, even for a beneficial ratio $\lambda _\varTheta /\lambda _u\approx 1$ (figure 5e,f; grey symbols). When the rotation-controlled regime begins at $\lambda _\varTheta /\lambda _u\geqslant 1$, Ekman pumping is immediately most effective, resulting in a sharp increase of the heat transport, which afterwards decreases again with further increase of $\lambda _\varTheta /\lambda _u$ (figure 5b,c). Heat transport enhancement does still exist even when the rotation-controlled regime begins at $\lambda _{\varTheta }/\lambda _u\gg 1$, but may not suffice to exceed the reference heat transport $ {\textit {Nu}}_0$ (figure 5c,f; $\varGamma ^{-1}=10$). These observations show that Ekman pumping is the essentially required mechanism for heat transport enhancement, and that the combination with a boundary layer ratio $\lambda _\varTheta /\lambda _u\approx 1$ is the beneficial condition.

Ekman pumping is always present in all rotating (sub-)regimes (figure 1b; blue, green, red areas). However, its net contribution to heat transport enhancement depends on the boundary layer ratio $\lambda _\varTheta /\lambda _u\approx 1$, which determines the efficiency of heat injection into the Ekman vortices (Stevens et al. Reference Stevens, Clercx and Lohse2010b). With increasing rotation vertical (heat) transport is suppressed, which leads to a thicker thermal boundary layer and hence a larger ratio $\lambda _\varTheta /\lambda _u$. Accordingly, the envelope of the normalized heat transport among all $\varGamma ^{-1}$ decreases with increasing boundary layer ratio for $\lambda _{\varTheta }/\lambda _u>1$ (figure 5). Another way to grasp this limiting effect is the intrinsic relation of $ {\textit {Nu}}/ {\textit {Nu}}_0$ and $\lambda _{\varTheta }/\lambda _u$ via the thermal shortcut. However, for the same boundary layer ratio $\lambda _\varTheta /\lambda _u$, i.e. for the same efficiency for heat being injected into the Ekman vortices, the heat transport is mostly largest whenever a stable, domain-spanning flow structure (single vortex or double vortex) is formed (figure 5d–f; green and red vs blue symbols). Moreover, single-vortex flow is less efficient than double-vortex flow for $ {\textit {Ra}}=7\times 10^{8}$ (figure 5e,f; green vs red symbols). Taken together, the heat transport with double-vortex or single-vortex flow clusters above the heat transport without a specific vortex structure, but within all regimes the heat transport decreases with increasing $\lambda _\varTheta /\lambda _u>1$. This nicely depicts the superposing impacts of the boundary layer ratio and a stable, domain-spanning flow formation (figure 5d–f).

In summary, we identified two effects that affect the efficiency of Ekman pumping in the rotation-controlled regime. Both effects can be characterized differently. First, the boundary layer ratio $\lambda _\varTheta /\lambda _u$ determines how effective heat is injected from the boundary layer into the bulk. Therefore, we refer to it as a heat injection effect. Second, certain combinations of confinement $\varGamma ^{-1}$ and rotation $ {\textit {Ro}}^{-1}$ lead to stable and vertically domain-spanning flow structures, which increase the efficiency of heat transported through the bulk. It is, therefore, a bulk transport effect. Several sub-regimes of stable, domain-spanning flow, which interfere with the effectiveness of the boundary layer ratio, result in the multiple maxima of heat transport under rotation and confinement.

5. Dependence on the Rayleigh number

We will now demonstrate how the three heat transport maxima (§ 3) depend on $ {\textit {Ra}}$. Therefore, we explore the $( {\textit {Ro}}^{-1},\varGamma ^{-1})$ parameter space at four values of $ {\textit {Ra}}=2\times 10^{8}$, $7\times 10^{8}$, $2.3\times 10^{9}$ and $7\times 10^{9}$ (figure 6). For each $ {\textit {Ra}}$ we can identify the different regimes of flow motion with their associated heat transport maxima (confinement, double vortex and single vortex). The confinement maximum is clearly present for all $ {\textit {Ra}}$, and always disappears when rotation is added (figure 6). The magnitude of heat transport enhancement increases with increasing $ {\textit {Ra}}$ (figure 7a), and shifts towards stronger confinement (figure 7c) as also observed by Chong et al. (Reference Chong, Huang, Kaczorowski and Xia2015) and Hartmann et al. (Reference Hartmann, Chong, Stevens, Verzicco and Lohse2021).

Figure 6. Heat transport (a,c,e,g) and regimes of prominent flow motion (b,d,f,h) in the parameter space of rotation rate $ {\textit {Ro}}^{-1}$ and cylinder confinement $\varGamma ^{-1}$ at four $ {\textit {Ra}}$. A, B and C mark the position of the confinement (§ 3.1), double-vortex (§ 3.2) and single-vortex maxima (§ 3.3), respectively. Grey, green and red lines show the transitions between the (sub-)regimes of prominent flow motion. The grey area depicts the buoyancy-dominated regime ($\textit {Re}_z>\textit {Re}_H$). The coloured areas belong to the rotation-controlled regime ($\textit {Re}_H>\textit {Re}_z$). In the green and red sub-regimes the characteristic flow motion indicates the double-vortex and single-vortex flow, respectively. The symbol colour always shows the heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$. $ {\textit {Nu}}_0$ is the heat transport of the non-rotating $\varGamma ^{-1}=1$ case at each $ {\textit {Ra}}$. See also figure 1(a,b) for detailed description.

Figure 7. (a) Normalized heat transport $ {\textit {Nu}}_{{max}}/ {\textit {Nu}}_0$ of the three maxima as a function of $ {\textit {Ra}}$. Here, $ {\textit {Nu}}_0$ is the heat transport of the non-rotating $\varGamma ^{-1}=1$ case at each $ {\textit {Ra}}$. (b) Rotation rate $ {\textit {Ro}}^{-1}_{opt}$ of the double-vortex and single-vortex maxima as a function of $ {\textit {Ra}}$. (c) Confinement parameter $\varGamma ^{-1}_{opt}$ of the three maxima as a function of $ {\textit {Ra}}$.

The double-vortex maximum is present at all four $ {\textit {Ra}}$, but with a sharp drop of its enhancement efficiency between $ {\textit {Ra}}=7\times 10^{8}$ and $ {\textit {Ra}}=2.3\times 10^{9}$. For the lower $ {\textit {Ra}}=2\times 10^{8},7\times 10^{8}$, the double-vortex maximum achieves the largest enhancements within this study by up to $50\;\%$, whereas the enhancement drops to less than $20\;\%$ for larger $ {\textit {Ra}}=2.3\times 10^{9},7\times 10^{9}$, which is even below the heat transport of the confinement maximum that then becomes the most efficient (figure 7a). Although the magnitude of heat transport enhancement decreases, the region of enhanced heat transport around the double-vortex maximum (where $ {\textit {Nu}}> {\textit {Nu}}_0$) enlarges significantly with increasing $ {\textit {Ra}}$. Most prominently the enhancement region enlarges towards larger $ {\textit {Ro}}^{-1}$ and $\varGamma ^{-1}$, concurrent with the shift of the associated flow regimes (figure 6). This entire behaviour of the heat transport enhancement can be explained by an interference of the two enhancing effects, beneficial boundary layer ratio and stable vortex formation.

To illustrate this interference that explains the above behaviour (figures 6, 7a), we show again the heat transport mapped onto the boundary layer ratio (as in figure 5), but now for selected confinement parameters $\varGamma ^{-1}$ among all four $ {\textit {Ra}}$ values (figure 8). Note that, in this plot, we assign each combination of $\varGamma ^{-1}$ and $ {\textit {Ra}}$ relative to its associated optimal confinement of the double-vortex maximum $\varGamma ^{-1}_{2VM}( {\textit {Ra}})$, i.e. $\varGamma ^{-1}=6>\varGamma ^{-1}_{2VM}( {\textit {Ra}}=2\times 10^{8})=4$ (figure 8c), but $\varGamma ^{-1}=6<\varGamma ^{-1}_{2VM}(7\times 10^{9})=8$ (figure 8a). The strong enhancement at lower $ {\textit {Ra}}$ results from the coincidence that, around the most beneficial boundary layer ratio $\lambda _\varTheta /\lambda _u\approx 1$, the flow (i) experiences Ekman pumping, and (ii) forms a stable double-vortex structure (figures 8b,e and 8a,d). Consequently, both enhancing effects interfere most efficiently, but also most locally around $\lambda _\varTheta /\lambda _u=1$ and in the $( {\textit {Ro}}^{-1},\varGamma ^{-1})$ parameter space. On the other hand, at larger $ {\textit {Ra}}$, the rotation-controlled regime begins only at $\lambda _\varTheta /\lambda _u\gtrsim 1$, which already reduces the largest achievable efficiency of Ekman pumping. Moreover, double-vortex flow is only formed at larger $\lambda _\varTheta /\lambda _u$, and thus it co-occurs only with a weak efficiency from the boundary layer ratio. Hence, the double-vortex maximum forms, where the interference of both effects is still the largest, but not as ideal as at lower $ {\textit {Ra}}$. As a consequence, double-vortex flow at its maximum is less stable at larger $ {\textit {Ra}}$ than at lower $ {\textit {Ra}}$ (figure 9). Still, whenever double-vortex flow can be identified, the heat transport is temporary strongly enhanced (figure 9; shaded areas). In brief, the shift of the double-vortex flow regime away from $\lambda _\varTheta /\lambda _u\approx 1$ for larger $ {\textit {Ra}}$ reduces the interference of the two enhancing effects, and therefore enlarges the region of heat transport enhancement while flattening the double-vortex maximum.

Figure 8. Normalized heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$ mapped onto the ratio of thermal and kinetic boundary layers $\lambda _\varTheta /\lambda _u$ for various combinations of $\varGamma ^{-1}$ and $ {\textit {Ra}}$: (b) for $\varGamma ^{-1}_{2VM}( {\textit {Ra}})$, where the double-vortex maximum is observed at each value of $ {\textit {Ra}}$, (a) for less confinement $\varGamma ^{-1}<\varGamma ^{-1}_{2VM}( {\textit {Ra}})$, i.e. wider cylinders than in (b), (c) for more confinement $\varGamma ^{-1}>\varGamma ^{-1}_{2VM}( {\textit {Ra}})$, i.e. more  slender cylinders than in (b). (d–f) Same data as in (a–c), respectively, but symbols coloured in their corresponding regime of prominent flow motion (figure 6): grey – buoyancy-dominated regime, blue – rotation-controlled regime, green – rotation-controlled with double-vortex flow, red – rotation-controlled with single-vortex flow. The vertical grey line marks the most beneficial boundary layer ratio $\lambda _\varTheta /\lambda _u=1$. The dashed grey line (no specific scaling) serves as guide for the eye. The coloured and grey lines connecting the data points in (a–c) and (d–f), respectively, follow increasing rotation $ {\textit {Ro}}^{-1}$ per confinement $\varGamma ^{-1}$.

Figure 9. Flow characteristics at the double-vortex maximum at $ {\textit {Ra}}=2\times 10^{8}$ and $ {\textit {Ra}}=2.3\times 10^{9}$: (a) temporal evolution of $ {\textit {Nu}}(t)$ at the top and bottom plates, (b) temporal evolution of $\textit {Re}(t)$ for each velocity component. The shaded areas indicate when double-vortex flow (as in figure 3a) is observed.

For very large $ {\textit {Ra}}$, this shift could result in a segregation of the double-vortex maximum into two separate maxima, one originating from the most effective boundary layer ratio $\lambda _\varTheta /\lambda _u\approx 1$, and another from stable double-vortex flow. Some evidence for such a splitting might be visible in our data for $ {\textit {Ra}}=7\times 10^{9}$ (figure 6g,h). This would be analogue to the separation of the single-vortex maximum from the double-vortex maximum with increasing $ {\textit {Ra}}$ (figure 6). To verify this hypothesis, either a much finer sampling in $ {\textit {Ro}}^{-1}$ and $\varGamma ^{-1}$ or another set at a larger $ {\textit {Ra}}$ is necessary, which both are beyond the computational resources of this study.

The single-vortex maximum emerges with increasing $ {\textit {Ra}}$ (figure 6). The regime of single-vortex flow is still very limited for $ {\textit {Ra}}=2\times 10^{8}$ and enlarges significantly with increasing Ra. For $ {\textit {Ra}}\geqslant 7\times 10^{8}$ the sub-regime of single-vortex flow has sufficiently enlarged to form an individual maximum within our parameter sampling. Still, we can define the single-vortex maximum for $ {\textit {Ra}}=2\times 10^{8}$ based on the location of the most stable single-vortex flow at $\varGamma ^{-1}=6$, $ {\textit {Ro}}^{-1}\!\approx\! 16$ (Appendix B: figure 14c). However, it seems likely that the single-vortex flow is associated with larger $ {\textit {Ra}}$, and does not occur at very low $ {\textit {Ra}}\ll 10^{8}$. With increasing $ {\textit {Ra}}$ and an enlarging single-vortex regime the enhancement efficiency of its maximum increases up to ${\approx }20\,\%$, which is comparable to the double-vortex maximum at the larger $ {\textit {Ra}}$ (figure 7a). Independent of $ {\textit {Ra}}$, the single-vortex flow is limited to relatively strong confinement $\varGamma ^{-1}>\varGamma ^{-1}_{2VM}$, and occurs only for $\lambda _\varTheta /\lambda _u\!>\!1$ directly after the transition to the rotation-controlled regime (figure 8c,f). As shown before (§ 3.3, Appendix B: figure 11), this transition occurs gradually with extending periods in which the flow stays in a stable single-vortex state. Both double-vortex and single-vortex maxima shift towards faster rotation and stronger confinement for increasing $ {\textit {Ra}}$ (figure 7b,c). Presumably, the locations of the maxima follow some effective scaling laws, but due to the large uncertainty we refrain from giving estimates for the scaling exponents.

The general trend of heat transport enhancement in wider vs slender cylinders, as already observed for $ {\textit {Ra}}=7\times 10^{8}$ (§ 4, figure 5), is present among all four $ {\textit {Ra}}$ (figure 8a–c): in wider cylinders ($\varGamma ^{-1}<\varGamma ^{-1}_{2VM}( {\textit {Ra}}$)), the rotation-controlled regime (blue symbols) begins at $\lambda _\varTheta /\lambda _u\ll 1$, and we observe a nearly symmetric heat transport enhancement around $\lambda _{\varTheta }/\lambda _u=1$ (figure 8a,d). For $\varGamma ^{-1}_{2VM}( {\textit {Ra}})$, the onset of heat transport enhancement occurs at $\lambda _{\varTheta }/\lambda _u\approx 1$ and can be rather steep because the flow can immediately form a double-vortex structure as observed for lower $ {\textit {Ra}}$ (figure 8b,e). In slender cylinders ($\varGamma ^{-1}>\varGamma ^{-1}_{2VM}( {\textit {Ra}}$)), the heat transport enhancement always shows a sharp onset at $\lambda _{\varTheta }/\lambda _u>1$, which corresponds to an immediate transition from buoyancy-dominated regime to rotation-controlled regime with single-vortex flow (figure 8c,f). This representation of our data nicely depicts a $ {\textit {Ra}}$-independent clustering of the heat transport within the different flow regimes (figure 8d–f).

Collapsing all data into one frame emphasizes even better a sheet-like clustering (figure 10), which nicely illustrates the superposing character of both enhancing effects. In the buoyancy-dominated regime (grey symbols), no heat transport enhancement by rotation is observed among all combinations of $\varGamma ^{-1}$, $ {\textit {Ro}}^{-1}$ and $ {\textit {Ra}}$, and all enhancement is related to the confinement maximum, which collapses at $\lambda _{\varTheta }/\lambda _u\approx 0.6$ for all $ {\textit {Ra}}$ (see also § 4). Only when the azimuthal motion becomes predominant does Ekman pumping set in and enhance the heat transport (blue symbols). Thereby, the net enhancement and the resulting heat transport depend on the actual ratio between the kinetic and thermal boundary layer thicknesses. When additionally a double-vortex or single-vortex flow is present, the heat transport is further increased (green and red symbols) above the cases when only Ekman pumping is present. Thereby, single-vortex flow seems to be very limited to a much smaller interval of $\lambda _{\varTheta }/\lambda _u$ than double-vortex flow. Whether Ekman pumping is active, and whether a (stable) domain-spanning flow is formed in addition depends on the combination of $ {\textit {Ra}}$, $\varGamma ^{-1}$, $ {\textit {Ro}}^{-1}$ and likely $\textit {Pr}$ that was not varied in this study. In other words: although a $ {\textit {Ra}}$-independent clustering is observed, several parameter values (e.g. larger $ {\textit {Ra}}$) can inhibit the best interference of a beneficial boundary layer ratio and a stable vortex flow, such that the largest potential heat transport enhancement is not achievable.

Figure 10. Clustering of the heat transport $ {\textit {Nu}}/ {\textit {Nu}}_0$ mapped onto the ratio of thermal and kinetic boundary layers $\lambda _\varTheta /\lambda _u$ for all combinations of $ {\textit {Ro}}^{-1}$, $\varGamma ^{-1}$ and $ {\textit {Ra}}$. The data are coloured in the corresponding regimes of flow motion (figure 6): grey – without Ekman pumping, blue – with Ekman pumping, green and red – Ekman pumping and double-vortex or single-vortex flow. The vertical grey line marks the most beneficial boundary layer ratio $\lambda _\varTheta /\lambda _u=1$. The dashed grey line (no specific scaling) serves as guide for the eye. The grey lines connecting the data points follow increasing rotation $ {\textit {Ro}}^{-1}$ per confinement $\varGamma ^{-1}$.

For all $ {\textit {Ra}}$, the onset of heat transport enhancement is nicely described by the transition of predominant fluid motion from vertical to horizontal $\textit {Re}_z=\textit {Re}_H$ (figures 6, 8). This transition and the onset of heat transport enhancement clearly depend on $ {\textit {Ra}}$, in contrast to describing the onset by a $ {\textit {Ra}}$-independent Ginzburg–Landau model (Weiss et al. Reference Weiss, Stevens, Zhong, Clercx, Lohse and Ahlers2010; Weiss & Ahlers Reference Weiss and Ahlers2011a,Reference Weiss and Ahlersb). Nonetheless, a $ {\textit {Ra}}$-dependence in very confined systems seems meaningful regarding that the formation of horizontally dominated, vortical flow requires a sufficiently wide domain, larger than the most unstable convective length scale $L_c\propto {\textit {Ek}}^{1/3}\propto {\textit {Ra}}^{1/6}$. However, the $ {\textit {Ra}}$-dependence appears to be rather complex so that its complete derivation is beyond the scope of this study.