2. Simulation methodology

We solve the non-dimensional Oberbeck–Boussinesq equations

(2.1)$$\begin{gather} \frac{\partial {\boldsymbol u}}{\partial t} + {\boldsymbol u} \boldsymbol{\cdot}\boldsymbol{\nabla} {\boldsymbol u} ={-}\boldsymbol{\nabla} p + T \hat{z} - \frac{1}{Ro}\,\hat{z} \times {\boldsymbol u} + \sqrt{\frac{Pr}{Ra}}\,\nabla^2 {\boldsymbol u}, \end{gather}$$

(2.2)$$\begin{gather}\frac{\partial T}{\partial t} + {\boldsymbol u} \boldsymbol{\cdot} \boldsymbol{\nabla} T = \frac{1}{\sqrt{Pr\,Ra}}\,\nabla^2 T, \end{gather}$$

(2.3)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol u} = 0 , \end{gather}$$

where ${\boldsymbol u}\ (\equiv u_x \hat {x} + u_y \hat {y} + u_z \hat {z})$, $T$ and $p$ are the velocity, temperature and pressure fields, respectively. The normalizing length $H$ is the height between the horizontal plates, and $\Delta T$ is the temperature difference between them. The free-fall velocity $u_f = \sqrt {\alpha g\,\Delta T\,H}$ and the free-fall time $t_f = H/u_f$ are the relevant velocity and time scales. The Rayleigh number is $Ra = \alpha g\,\Delta T\,H^3/(\nu \kappa )$, and the Prandtl number is $Pr = \nu /\kappa$. The convective Rossby number $Ro = u_f/(2\varOmega H) = \alpha g\, \Delta T/(2\varOmega u_f)$ is the ratio of the buoyancy and Coriolis forces, where $\varOmega$ is the rotation rate, and $\alpha, \nu, \kappa$ are the isobaric coefficient of thermal expansion, kinematic viscosity and thermal diffusivity of the fluid, respectively.

The simulations correspond to $Pr = 0.021$ and $2 \times 10^7 \leq Ra \leq 10^{10}$ in a cylindrical cell with $\varGamma = 0.1$ using the solver Nek5000, based on the spectral element method (Fischer Reference Fischer1997). The no-slip boundary condition is prescribed for the velocity field on all walls, and the isothermal and adiabatic conditions for the temperature field on the horizontal and sidewalls, respectively. The cylinder is decomposed into $N_e$ elements, and the turbulence fields within each element are expanded using the $N$th-order Lagrangian interpolation polynomials. Thus the number of mesh cells in the entire flow is $N_e N^3$; higher mesh density in the near-wall regions is used to capture rapid variations of the field variables. More details can be found in Scheel, Emran & Schumacher (Reference Scheel, Emran and Schumacher2013), Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020) and Pandey et al. (Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a). (Incidentally, the number of spectral elements $N_e$ in Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020) was 192 000 for $Ra=10^8, 10^9, 10^{10}$ and $10^{11}$.)

The effects of rotation are studied using two different approaches. First, the effects of increasing thermal forcing are explored for a fixed $Ek = 1.45 \times 10^{-6}$ and varying $Ra$ up to $10^{10}$. The Ekman number $Ek = \nu /(2 \varOmega H^2)$ quantifies the strength of the viscous force relative to that of the Coriolis force, so we are dealing with a rapidly rotating case. Second, the effects of increasing rotation are studied by fixing $Ra = 10^8$, $10^9$ and $10^{10}$, and by decreasing the Rossby number for each $Ra$. Note that the convective Rossby number is also expressed as $Ro = Ek\,\sqrt {Ra/Pr}$; for a fixed $Ra$ and $Pr$, $Ek$ decreases with the decreasing $Ro$. The simulations for non-rotating convection serve as the reference state. To compare the flow properties with those of moderate-$Pr$ convection, we additionally conduct RRBC simulations for $Pr = 1$ and $Ra$ up to $10^{11}$, but the emphasis in this paper is the low-$Pr$ case. The parameter space in this study is shown in figure 1.

Figure 1. The parameter space explored in the present study for (a) $Pr = 0.021$ and (b) $Pr = 1$. Open symbols are for simulations with fixed rotation and varying thermal forcing, whereas filled ones are for simulations with fixed forcing and varying rotation rate. In (a,b), the sloping data are for variable $Ro$ but constant $Ek$.

The Kolmogorov length scale is estimated as $\eta = (\nu ^3/\varepsilon _u)^{1/4}$, where $\varepsilon _u$ is the kinetic energy dissipation rate computed at each point in the flow as

(2.4)\begin{equation} \varepsilon_u({\boldsymbol x}) = \frac{\nu}{2} \sum_{i,j} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)^2, \end{equation}

where $i,j \equiv (x,y,z)$. To ensure the adequacy of the spatial resolution, we estimate the height-dependent Kolmogorov scale $\eta (z)$ using the area- and time-averaged dissipation rate $\langle \varepsilon _u \rangle _{A,t}(z)$, and ensure that the vertical grid spacing $\varDelta _z$ remains of the order $\eta (z)$. This constraint captures all significant variations in the velocity field. Further, within the Ekman layer, which varies as $\delta_u \sim \sqrt {Ek}$, we have embedded 5–20 grid points.

We briefly expand here on the computational gains in using a slender cell because the fluid volume is smaller by a factor of $\varGamma ^2$. Higher $Ra$ could thus be achieved for the same computational resources, compared to those of higher $\varGamma$. However, an increased fraction of fluid is affected by the sidewall, and the critical $Ra$ for the onset of convection grows for small $\varGamma$ (Shishkina Reference Shishkina2021; Ahlers et al. Reference Ahlers2022). To that extent, the computational advantage of using a slender domain to explore a highly turbulent regime of convection tends to be diminished, but one needs further exploration on these advantages in different Rayleigh number regimes.

3. Flow morphology

Multiple vertically stacked circulation rolls lead to helical structures in slender convection domains (Iyer et al. Reference Iyer, Scheel, Schumacher and Sreenivasan2020; Zwirner, Tilgner & Shishkina Reference Zwirner, Tilgner and Shishkina2020; Pandey & Sreenivasan Reference Pandey and Sreenivasan2021; Pandey et al. Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a). The flow configuration in the non-rotating slender cell is shown in figure 2 for varying $Ra$. The instantaneous velocity streamlines shown in figures 2(a–c), coloured according to the vertical velocity, confirm the presence of vertically stacked rolls. The helical flow structure is relatively smooth for $Ra = 10^8$ (figure 2a) but becomes increasingly complex as the thermal forcing increases. The vertical velocity slices in figures 2(d–f) exhibit coherently moving flows, both up and down, with sizes comparable to the lateral extent of the flow. However, these organized structures incorporate increasingly smaller scales as $Ra$ increases. The corresponding temperature isosurfaces in figures 2(g–i) show that the mixing is weak at low $Ra$ but becomes increasingly effective as the thermal forcing becomes stronger. Even in a highly turbulent flow for $Ra = 10^{10}$ (figure 2i), a variety of temperature isosurfaces are present in the bulk region, which indicates that the turbulent mixing is weaker than in wider convection domains, where a well-mixed and isothermal bulk component is observed. The global heat transfer, however, is not very different in the two cases (Pandey et al. Reference Pandey, Krasnov, Sreenivasan and Schumacher2022b).

Figure 2. Instantaneous convective structures in a non-rotating slender cell for (a,d,g) $Pr = 0.021$ and $Ra = 10^8$, (b,e,h) $Ra = 10^9$, and (c,f,i) $Ra = 10^{10}$. The velocity streamlines (a–c), coloured by the vertical velocity, exhibit helical flow structures in the slender cell. Planar cuts of the vertical velocity (d–f) reveal that progressively finer flow structures are generated with increasing thermal forcing. Isosurfaces of the temperature (g–i) indicate that despite increased mixing with $Ra$, the isothermal bulk region, observed to exist in wider convection domains, is not present in the slender cell.

The critical parameters for the onset of non-rotating convection are independent of $Pr$. In contrast, the onset parameters in rotating convection do depend on the Prandtl number when it is less than 0.68 (Chandrasekhar Reference Chandrasekhar1981). Linear stability analysis for $Pr > 0.68$ in horizontally unconfined domains yields the Rayleigh number and the length scale for the steady onset as

(3.1)$$\begin{gather} Ra_c = 3 ({\rm \pi}^2/2)^{2/3}\,Ek^{{-}4/3} \approx 8.7\,Ek^{{-}4/3}, \end{gather}$$

(3.2)$$\begin{gather}\ell_c/H = (2{\rm \pi}^4)^{1/6}\,Ek^{1/3} \approx 2.4\,Ek^{1/3}. \end{gather}$$

For low Prandtl numbers ($Pr < 0.68$), the critical parameters at the oscillatory onset depend on both $Ek$ and $Pr$ (Horn & Schmid Reference Horn and Schmid2017; Aurnou et al. Reference Aurnou, Bertin, Grannan, Horn and Vogt2018; Vogt, Horn & Aurnou Reference Vogt, Horn and Aurnou2021) as

(3.3)$$\begin{gather} Ra_c = 3 {\rm \pi}\left(\frac{2 {\rm \pi}}{1+Pr} \right)^{1/3} \left( \frac{Ek}{Pr} \right)^{{-}4/3}, \end{gather}$$

(3.4)$$\begin{gather}\ell_c/H = (2 {\rm \pi}^4)^{1/6} (1+Pr)^{1/3} \left( \frac{Ek}{Pr} \right)^{1/3}, \end{gather}$$

(3.5)$$\begin{gather}\omega_c = (2 - 3\,Pr^2)^{1/2} \left( \frac{2 {\rm \pi}}{1+Pr} \right)^{2/3} \left( \frac{Ek}{Pr} \right)^{1/3}, \end{gather}$$

where $\omega_c$ is the oscillation frequency at the onset. Thus the onset length scale $\ell _c$ in low-$Pr$ convection is larger by a factor of $(1+1/Pr)^{1/3}$.

The flow in the rotating cell for $Ek = 1.45 \times 10^{-6}$ is shown in figure 3 for various $Ra$. For this $Ek$ and $Pr = 0.021$, the length scale at the onset of convection according to (3.4) is $\ell _c/H \approx 0.1$, which is equal to the horizontal dimension of the slender domain. Thus the low-$Pr$ flow for this $Ek$ is confined at the onset. For $Ra = 6 \times 10^7$ – not far from the onset – the Coriolis force dominates the buoyancy force, leading to smooth and tall velocity structures inhabiting the entire depth (figure 3a). For $Ra = 2 \times 10^8$ (figure 3b), buoyancy becomes stronger but the flow continues to be influenced by the strong rotation. The observed tall structures develop a wavy character as in high-$Pr$ convection (Cheng et al. Reference Cheng, Madonia, Aguirre Guzmán and Kunnen2020). The vertical coherence is lost nearly completely for $Ra = 10^9$, and for $Ra = 10^{10}$, the flow morphology appears very close to that in the non-rotating cell shown in figures 2(f,i), indicating that the effects of the Coriolis force (for this rotation) essentially vanish near $Ra = 10^{10}$. From the velocity streamlines visualization (not shown), we infer that the helical structure, present for the entire range of the thermal forcing explored in the non-rotating cell, is not observed in the rotating slender convection when the Coriolis force dominates; the helical configuration is recovered only when the thermal forcing becomes strong enough to overcome rotation.

Figure 3. Flow morphology in a rotating slender cell for $Pr = 0.021$, $Ek = 1.45 \times 10^{-6}$, revealed by (a–d) instantaneous vertical velocity slices and (e–h) temperature isosurfaces, for (a,e) $Ra = 6 \times 10^7$, (b,f) $Ra = 2 \times 10^8$, (c,g) $Ra = 10^9$, and (d,h) $Ra = 10^{10}$. Near the onset of convection (a,e), flow structures feel the rotation strongly, and the variation along the vertical direction is almost suppressed. With increasing $Ra$, the resilience increases and the flow configuration for $Ra = 10^{10}$ (d,h) shows strong resemblance with its non-rotating counterpart in figure 2. (The global heat and momentum transports are also nearly indifferent for these cases; see table 3.)

Dwindling vertical coherence with increasing $Ra$, for a fixed rotation, is also clear from the temperature field in figures 3(e–h). The temperature isosurfaces for $Ra = 6 \times 10^7$ – shown in figure 3(e) – are nearly flat circular discs. This is in line with the Taylor–Proudman constraint that the vertical variation of the flow is inhibited in a rapidly rotating inviscid flow (Chandrasekhar Reference Chandrasekhar1981). With increasing $Ra$, the isosurfaces become increasingly three-dimensional, and for $Ra = 10^{10}$, appear very similar to the non-rotating case. In § 5, we also show that the integral transport properties of the rotating flow at $Ra = 10^{10}$ and $Ek = 1.45 \times 10^{-6}$ are nearly the same as those of the corresponding non-rotating flow.

A qualitatively similar change in the flow morphology is observed when the rotation increases for a prescribed thermal forcing (Horn & Shishkina Reference Horn and Shishkina2015; Aurnou et al. Reference Aurnou, Horn and Julien2020). Figure 4 exhibits flow structures for $Ra = 10^{10}$ and $0 \leq Ro^{-1} \leq 30$, where the helical structure transforms to tall vertically elongated velocity structures as the container is rotated increasingly rapidly. The temperature contours also lose their three-dimensional character as $Ro$ decreases, consistent with the observations in wider convection domains filled with moderate- and high-$Pr$ fluids (Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015). We also observe from figure 4 that the flow length scale varies with varying $Ro$. For $Ro^{-1} = 30$ in figure 4(d), $Ek \approx 4.8 \times 10^{-8}$ and the linear stability theory yields $\ell _c/H \approx 0.032$, which is nearly three times smaller than that for $Ek \approx 1.45 \times 10^{-6}$ in figure 3. Therefore, the confinement effects in the slender cell are mitigated progressively as $Ek$ decreases.

Figure 4. Evolution of the convective structures with increasing rotation rate for $Pr = 0.021$, $Ra = 10^{10}$: (a,e) $Ro^{-1} = 0$, (b,f) $Ro^{-1} = 2$, (c,g) $Ro^{-1} = 10$, (d,h) $Ro^{-1} = 30$. The flow loses its three-dimensional character, and the length scale of the velocity structures decreases, as the Rossby number decreases.

4. Rotating slender convection near the onset

We first explore the flow evolution in the low-supercritical regime for $Ek = 1.45 \times 10^{-6}$ and $Pr = 0.021$. We start simulations from the conduction solution with random perturbations, and observe that the convective state, corresponding to substantially non-zero values of $Nu-1$, occurs first at $Ra = 5.6 \times 10^7$. Note that this value is nearly an order of magnitude larger than the $Ra_c$ obtained from (3.3). Here, the Nusselt number $Nu$ is computed as

(4.1)\begin{equation} Nu = 1 + \sqrt{Ra\,Pr} \, \langle u_z T \rangle_{V,t}, \end{equation}

where $\langle \cdot \rangle _{V,t}$ denotes averaging over the entire flow and integration time. For a simulation at $Ra = 5.55 \times 10^7$ started from the conduction state, we observe $Nu - 1 \approx 0.0022$, whereas we get $Nu -1 \approx 0.049$ when the same simulation is started with a flow state given by the simulation at $Ra = 6 \times 10^7$. By decreasing $Ra$, we can observe finite-amplitude convection up to $Ra = 5.40 \times 10^7$, where the convective flux $Nu - 1 \approx 0.034$ is small but significantly different from zero. Thus there is modest hysteresis in low-$Pr$ RRBC in a slender cell.

We monitor the evolution of the temperature and velocity fields at a few locations in the flow, and show the temperature variation at mid-height near the sidewall in figure 5 for $Ra \leq 10^8$ and $Ek = 1.45 \times 10^{-6}$. Figure 5(a) exhibits that the flow evolves periodically for $Ra = 6 \times 10^7$, a feature also observed for lower-$Ra$ simulations. The corresponding power spectrum, shown in figure 5(b), reveals a single dominant frequency at $\omega \approx 0.20$, and its higher harmonics. It is interesting that this frequency agrees well with $\omega _c \approx 0.195$ predicted from (3.5) using the linear stability analysis at $Ek = 1.45 \times 10^{-6}$ (Chandrasekhar Reference Chandrasekhar1981). With increasing $Ra$, the flow evolution becomes progressively complex due to the emergence of other modes. For $Ra = 7 \times 10^7$, a high-amplitude peak develops also at a lower frequency, which indicates the presence of the wall modes (Goldstein et al. Reference Goldstein, Knobloch, Mercader and Net1994; Horn & Schmid Reference Horn and Schmid2017; Aurnou et al. Reference Aurnou, Bertin, Grannan, Horn and Vogt2018). For $Ra = 8 \times 10^7$, the peak at lower frequency becomes strong compared to its higher-frequency counterpart. The periodicity is nearly lost at $Ra = 9 \times 10^7$, and the flow becomes chaotic. The broadband power spectrum for $Ra \geq 9 \times 10^7$ indicates the presence of flow structures of a wide range of temporal (and spatial) scales. Thus, due to its highly inertial nature, low-$Pr$ RRBC becomes promptly complex.

Figure 5. (a,c,e,g,i) Temperature signal in the mid-plane at a probe near the sidewall, and (b,d,f,h,j) the corresponding power spectrum in a rapidly rotating flow ($Pr = 0.021$, $Ek = 1.45 \times 10^{-6}$) near the onset of convection: (a,b) $Ra = 6 \times 10^7$, (c,d) $Ra = 7 \times 10^7$, (e,f) $Ra = 8 \times 10^7$, (g,h) $Ra = 9 \times 10^7$, and (i,j) $Ra = 10^8$.

Figure 6 shows the instantaneous mid-plane slices of the vertical velocity for $Ra \leq 9 \times 10^7$ in the rotating cell at $Ek = 1.45 \times 10^{-6}$. We can see that the vertical velocity peaks near the sidewall at $Ra = 6 \times 10^7$, while the bulk region (away from the sidewall) is characterized by low-amplitude structures. This is a signature of the wall modes in the slender convection cell at a low Prandtl number (Horn & Schmid Reference Horn and Schmid2017; Aurnou et al. Reference Aurnou, Bertin, Grannan, Horn and Vogt2018). For $Ra = 7 \times 10^7$ and $8 \times 10^7$, the high-amplitude patch broadens and encroaches into the bulk interior. However, the interior is nearly entirely occupied by the bulk mode at $Ra = 9 \times 10^7$, and has taken over the wall modes (Goldstein et al. Reference Goldstein, Knobloch, Mercader and Net1994). Further, the convective flow patterns in rotating cylinders are observed to precess (mostly) in the retrograde direction (Zhong, Ecke & Steinberg Reference Zhong, Ecke and Steinberg1993; Horn & Schmid Reference Horn and Schmid2017). Similar precessing patterns in the slender cell at low Prandtl number can be found in the supplementary movies for a few cases available at https://doi.org/10.1017/jfm.2024.640.

Figure 6. Instantaneous vertical velocity contours in the mid-plane for $Ek = 1.45 \times 10^{-6}$ and (a) $Ra = 6 \times 10^7$, (b) $Ra = 7 \times 10^7$, (c) $Ra = 8 \times 10^7$, (d) $Ra = 9 \times 10^7$. Peak amplitudes in the velocity are observed near the sidewall at low $Ra$, but the interior of the domain is filled with stronger flows as thermal driving becomes stronger.

We now compare the heat transport at the onset in flows at low $Pr$ with those at moderate and high $Pr$, both rotating. For moderate and high $Pr$, the convective heat transport $Nu-1$ has been observed to increase linearly with the supercriticality $\epsilon = Ra/Ra_c - 1$ (Gillet & Jones Reference Gillet and Jones2006; Ecke Reference Ecke2015; Gastine, Wicht & Aubert Reference Gastine, Wicht and Aubert2016; Long et al. Reference Long, Mound, Davies and Tobias2020; Ecke et al. Reference Ecke, Zhang and Shishkina2022). Figure 7(a) shows the present data on $Nu-1$ as a function of $\epsilon$ on a linear–linear scale for $Pr = 0.021$ and $Ek = 1.45 \times 10^{-6}$. Even though there is modest hysteresis (as mentioned earlier), we have taken $Ra_c = 5.5 \times 10^7$ based on the observation that the convective heat transport is very small at $Ra = 5.55 \times 10^7$. Figure 7(a) shows a linear trend for $\epsilon \lesssim 0.5$, with the best fit given by $Nu-1 = 0.39\epsilon + 0.05$. The precise value of the finite intercept depends on the modest hysteresis just mentioned, so is probably not entirely reliable.

Figure 7. Convective heat transport $Nu-1$ as a function of the normalized distance $\epsilon = Ra/Ra_c-1$ from the onset for (a) $Pr = 0.021$ and (b) $Pr = 1$. Linear scaling is observed in the vicinity of the onset for both cases, but a finite intercept in (a) is due to the highly inertial nature of low-$Pr$ convection.

The data for unity Prandtl number in the same slender cell at a similar Ekman number, i.e. $Ek = 10^{-6}$, are shown in figure 7(b). For this case, the heat transport due to convective motion vanishes at $Ra \approx 8 \times 10^7$, this being the onset Rayleigh number. The data follow the linear scaling quite well; when extrapolated back to $Nu =1$, one obtains $Ra_c = 8 \times 10^7$, in perfect agreement with $Ra_c$ determined from inspecting the DNS. It is intriguing that (3.1) yields $Ra_c \approx 8.7 \times 10^8$, which is an order of magnitude higher than the $Ra_c$ determined from DNS data. This is due to wall modes that lower the critical Rayleigh number in confined domains (Herrmann & Busse Reference Herrmann and Busse1993; Aurnou et al. Reference Aurnou, Bertin, Grannan, Horn and Vogt2018; Vogt et al. Reference Vogt, Horn and Aurnou2021). Figure 7(b) further shows that the prefactor of the linear scaling is $\approx 2$, which is close to 1.54 reported recently in a $\varGamma = 1/2$ cell for $Pr = 0.8$ and $Ek = 10^{-6}$ (Ecke Reference Ecke2015; Ecke et al. Reference Ecke, Zhang and Shishkina2022). Thus a slightly different convective heat flux near the onset could be due to the highly inertial nature of low-$Pr$ convection, where the chaotic time dependence is ingrained even at the onset. It appears fair to conclude, overall, that the onset behaviour is essentially the same for all Prandtl numbers.

5. Global transport of heat and momentum in the turbulent state

We now compare the heat transport over an extended range of $Ra$ between rotating and non-rotating cases, both at low $Pr = 0.021$; see figure 8. The data for the non-rotating slender cell (green stars) do not follow a satisfactory power law, but we proceed to fit power laws for different segments of $Ra$, and comment on them. Let us first note that the critical Rayleigh number for the onset of convection in the slender cell is nearly $1.1 \times 10^7$ (Pandey & Sreenivasan Reference Pandey and Sreenivasan2021), which is much higher than that in unbounded domains (Chandrasekhar Reference Chandrasekhar1981). However, the temperature and velocity evolution in the flow as well as the averaged heat flux at the horizontal plates exhibit chaotic time dependence already for $Ra = 2 \times 10^7$. This indicates that the transition to turbulence in the slender cell for $Pr = 0.021$ occurs not far from the onset $Ra$, which is in line with the observations in wider domains (Schumacher, Götzfried & Scheel Reference Schumacher, Götzfried and Scheel2015; Horn & Schmid Reference Horn and Schmid2017). Figure 8 also plots the heat transport from non-rotating convection experiments by Glazier et al. (Reference Glazier, Segawa, Naert and Sano1999) in a $\varGamma = 1/2$ cell, and from DNS by Scheel & Schumacher (Reference Scheel and Schumacher2017) in a $\varGamma = 1$ cell, both at $Pr \approx 0.021$. While the heat transport in the slender cell is lower than that reported in wider cells, the discrepancy decreases with increasing $Ra$; the slender data at the largest $Ra$ explored in this work follow a scaling similar to that in wider convection cells.

Figure 8. Convective heat transport as a function of $Ra$ in the non-rotating slender cell (green stars) and in a rapidly rotating slender cell (red circles) of $\varGamma = 0.1$ for $Pr = 0.021$. Heat flux in the non-rotating cell exhibits a steeper scaling $Nu-1 \sim Ra^{1.03}$ (dashed green line) compared to that observed in wider convection cells for moderate Rayleigh numbers, but a similar $Ra^{1/3}$ scaling for large Rayleigh numbers (solid green line). The $Nu$ in rotating convection is lower than in non-rotating convection when Rayleigh numbers are small, but the differences essentially diminish as the thermal forcing increases. The data for $10^8 < Ra \leq 10^9$ exhibit a power law, and the best fit yields $Nu-1 \sim Ra^{1.32}$ (dashed red line), which is close to $Nu-1 \sim Ra^{3/2}$ scaling in the geostrophic regime. Cyan diamonds represent experimental data for $Ek = 10^{-6}$ in a $\varGamma = 1$ cylinder from King & Aurnou (Reference King and Aurnou2013), and the solid cyan line indicates $Ra^{1.32}$ scaling. Solid lines are not the best fits but are drawn as a guide to the eye. Filled symbols correspond to low-$Pr$ non-rotating convection from the literature: blue squares represent the experimental data from Glazier et al. (Reference Glazier, Segawa, Naert and Sano1999) in the $\varGamma = 1/2$ domain, whereas orange triangles correspond to DNS data in a $\varGamma = 1$ cell by Scheel & Schumacher (Reference Scheel and Schumacher2017).

It is well known that rotation reduces heat transport (Chandrasekhar Reference Chandrasekhar1981; Plumley & Julien Reference Plumley and Julien2019; Kunnen Reference Kunnen2021; Ecke & Shishkina Reference Ecke and Shishkina2023). The data for $Ek = 1.45 \times 10^{-6}$ (figure 8, red circles) confirm this behaviour – except for large $Ra \geq 10^9$, for which the Nusselt numbers in rotating and non-rotating cases are quite close, and for these particular conditions, $Nu$ can be said to be essentially unaffected by rotation, and the data nearly follow the canonical non-rotating $Ra^{1/3}$ scaling (Niemela et al. Reference Niemela, Skrbek, Sreenivasan and Donnelly2000) – indicating that the effects of rotation on the low-$Pr$ convection in the slender cell resemble those in wider domains.

Turning attention to low $Ra$ for, say, $Ra < 10^8$, $Nu$ is seen to follow a much steeper scaling, $Nu-1 \sim Ra^3$ (solid red line). (This does not contradict the linear scaling shown in figure 7(a), as these plots use different quantities.) This scaling regime is similar to that reported in DNS in a horizontally periodic box for $Pr = 1$ (Song, Shishkina & Zhu Reference Song, Shishkina and Zhu2024). Note that a steep heat transport scaling $Nu \sim Ra^3$ near the onset of rotating convection has been proposed by King et al. (Reference King, Stellmach and Aurnou2012), thus reported for moderate Prandtl numbers (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015). However, our $Nu$ versus $Ra$ plot (not shown) does not show this cubic scaling near the onset.

In the intermediate region $10^8 < Ra < 10^9$, the data for the rotating case seem to follow a power law with the best fit given by $Nu-1 \sim Ra^{1.32 \pm 0.06}$. This scaling is roughly consistent with simulations of the asymptotically reduced equations – describing RBC in the rapidly rotating limit – for which $Nu-1$ increases as $Ra^{3/2}$, for $Pr \geq 0.3$, in the geostrophic regime (Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a). A plot of $Nu$ versus $Ra$ (not shown) gives a lower exponent of 0.95 for the same range of $Ra$, which is similar to $Nu \sim Ra^{0.91}$ observed in the ‘rotationally dominated’ regime of convection in liquid gallium in a $\varGamma = 1.94$ cylinder (Aurnou et al. Reference Aurnou, Bertin, Grannan, Horn and Vogt2018). Moreover, $Nu-1 \sim Ra^{1.03}$ for intermediate Rayleigh numbers in the non-rotating case. For comparison, we also include data from King & Aurnou (Reference King and Aurnou2013), who performed experiments in a $\varGamma = 1$ cylindrical cell for $Pr \approx 0.025$: cyan diamonds in figure 8 show the heat transport for $Ek = 10^{-6}$. It is clear that $Nu-1$ in the wider RRBC cell also increases steeply near the onset, but for higher $Ra$, $Nu-1$ exhibits a similar $Ra^{1.32}$ scaling (solid cyan line) as observed in the rotating slender cell.

Rotation also influences momentum transport, as seen by the behaviour of the Reynolds number, based here on the root mean square (r.m.s.) velocity and the depth of the fluid layer, as

(5.1)\begin{equation} Re = \sqrt{\langle u_x^2+u_y^2+u_z^2 \rangle_{V,t}\,Ra/Pr}. \end{equation}

The Reynolds number in both the non-rotating and rotating cells is plotted as a function of $Ra$ in figure 9. The $Re$ for the non-rotating cell (green stars) increases rapidly near the onset, but the rate of increase decays as the thermal driving becomes stronger. In the same intermediate range of $Ra$ where we observe $Nu-1 \sim Ra$ scaling, the best fit yields $Re \sim Ra^{0.73 \pm 0.01}$ scaling. Note that the Reynolds number in wider convection domains has been known to increase nearly as $\sqrt {Ra}$ for moderate and low Prandtl numbers (Ahlers, Grossmann & Lohse Reference Ahlers, Grossmann and Lohse2009; Chillà & Schumacher Reference Chillà and Schumacher2012; Pandey & Verma Reference Pandey and Verma2016; Scheel & Schumacher Reference Scheel and Schumacher2017; Verma, Kumar & Pandey Reference Verma, Kumar and Pandey2017; Pandey et al. Reference Pandey, Krasnov, Sreenivasan and Schumacher2022b). We indicate the $Ra^{1/2}$ scaling by a solid green line in figure 9, and note that the non-rotating data for the largest few Rayleigh numbers of this study nearly follow this same scaling, signalling that the effects of confinement become weaker with increasing thermal driving. Also included in figure 9 for comparison are $Re$ computed in a $\varGamma = 1$ cylindrical cell for $Pr = 0.021$ by Scheel & Schumacher (Reference Scheel and Schumacher2017). We observe that the Reynolds number in the slender cell is smaller compared to that in the wider cell, which is due to a larger effective friction of rigid boundaries in the former case (Pandey & Sreenivasan Reference Pandey and Sreenivasan2021).

Figure 9. Reynolds number $Re$ as a function of $Ra$ in the non-rotating cell (green stars) and rotating cell at $Ek = 1.45 \times 10^{-6}$ (red circles). Velocity fluctuations grow rapidly near the onset of convection in the slender cell, but the growth rate becomes slower as the driving becomes stronger. The solid green line indicates that the data at the highest $Ra$ nearly follow a $\sqrt {Ra}$ power law as in wider cells. The solid red line suggests that $Re$ grows as $Ra^3$ for $Ra < 10^8$. Dashed lines represent the best fits for moderate thermal forcings. The difference between the non-rotating and rotating $Re$ values declines as $Ra$ increases, and the two are nearly indistinguishable at $Ra = 10^{10}$. Orange triangles represent DNS data in a $\varGamma = 1$ cell by Scheel & Schumacher (Reference Scheel and Schumacher2017).

Red circles in figure 9 represent the Reynolds numbers in the rotating slender cell for $Ek = 1.45 \times 10^{-6}$, and the reduced transport of momentum in the presence of rotation is clear (Schmitz & Tilgner Reference Schmitz and Tilgner2010) for low $Ra$. The figure also shows, similar to figure 8, that the onset of convection shifts to higher $Ra$ compared to that for the non-rotating cell. Near the convective onset, $Re$ in the rotating cell grows approximately as $Ra^3$ (solid red line), very similar to the growth of $Nu-1$ in this regime. A rapid growth of $Re$ near the onset of rotating convection was also reported by Schmitz & Tilgner (Reference Schmitz and Tilgner2010), who performed DNS in a horizontally periodic domain. With increase of the thermal driving, the growth rate of $Re$ decays, and the best fit for $10^8 < Ra \leq 10^9$ is a $Re \sim Ra^{0.83 \pm 0.03}$ scaling (dashed red line). This scaling has some similarity with the dissipation-free scaling $Re \sim Ra\,Ek/Pr$ reported by Guervilly, Cardin & Schaeffer (Reference Guervilly, Cardin and Schaeffer2019), Maffei et al. (Reference Maffei, Krouss, Julien and Calkins2021), Vogt et al. (Reference Vogt, Horn and Aurnou2021) and Ecke & Shishkina (Reference Ecke and Shishkina2023). For higher $Ra$, $Re$ in the rotating cell approaches that in the non-rotating cell, and the difference becomes very small for $Ra > 10^9$.

The influence of rotation can be studied also by decreasing the Rossby number $Ro$ for a fixed Rayleigh number (Kunnen et al. Reference Kunnen, Stevens, Overkamp, Sun, van Heijst and Clercx2011; Stevens, Clercx & Lohse Reference Stevens, Clercx and Lohse2013; Ecke & Niemela Reference Ecke and Niemela2014; Horn & Shishkina Reference Horn and Shishkina2015; Aurnou et al. Reference Aurnou, Horn and Julien2020); the inverse Rossby number $Ro^{-1}$ is a measure of the strength of the Coriolis force relative to buoyancy. We carry out low-$Pr$ simulations for $Ro^{-1} \in [0,30]$ at $Ra = 10^8, 10^9, 10^{10}$. The Nusselt number normalized with $Nu_0$ – the heat transport in absence of rotation – as a function of $Ro^{-1}$ is shown in figure 10(a), with the curves for different $Ra$ collapsing reasonably well. The normalized heat flux remains close to unity for $Ro^{-1} \lesssim 2$, beyond which it decreases. This indicates that slow rotation does not affect the heat transport in the slender cell, in line with observations in wider convection cells for moderate Prandtl numbers (Wedi et al. Reference Wedi, van Gils, Bodenschatz and Weiss2021). Figure 10 also shows that there is no enhancement of heat transport at moderate rotation rates, in contrast to that at large Prandtl numbers due to the so-called ‘Ekman pumping’ mechanism (Stevens et al. Reference Stevens, Zhong, Clercx, Ahlers and Lohse2009; Zhong et al. Reference Zhong, Stevens, Clercx, Verzicco, Lohse and Ahlers2009; Zhong & Ahlers Reference Zhong and Ahlers2010; Chong et al. Reference Chong, Yang, Huang, Zhong, Stevens, Verzicco, Lohse and Xia2017), but the absence of heat flux enhancement in the slender cell data is consistent with low-$Pr$ RRBC in more extended domains (Zhong et al. Reference Zhong, Stevens, Clercx, Verzicco, Lohse and Ahlers2009).

Figure 10. (a) Heat and (b) momentum transports in rotating slender cells, normalized with the corresponding values from the non-rotating cell, are nearly unity for $Ro^{-1} \leq 2$, but decay rapidly for larger inverse Rossby numbers. The suppression of the heat flux is stronger than that of the momentum flux in low-$Pr$ slender convection.

The normalized Reynolds number $Re/Re_0$ with $Ro^{-1}$ is plotted in figure 10(b). The trend is qualitatively similar to that of the normalized heat flux; weak rotation ($Ro^{-1} \lesssim 2$) does not affect momentum transport. For $Ro^{-1} \gtrsim 2$, the normalized momentum flux decreases, but the data for $Ra = 10^8$ lie below those for $Ra \geq 10^9$ at higher $Ro^{-1}$. The suppression of the momentum transport is weaker than for heat transport; at $Ro^{-1} = 10$, the normalized Nusselt number is $Nu/Nu_0 \approx 0.45$, whereas $Re/Re_0 \approx 0.8$, both for $Ra = 10^{10}$.

6. Temperature gradient in the bulk region and viscous boundary layer

The mean temperature in turbulent convection varies primarily in the thin thermal boundary layers near the horizontal plates. However, severe lateral confinement causes temperature variation to be present also in the bulk region for moderate and low Prandtl numbers (Iyer et al. Reference Iyer, Scheel, Schumacher and Sreenivasan2020; Pandey & Sreenivasan Reference Pandey and Sreenivasan2021; Pandey et al. Reference Pandey, Krasnov, Schumacher, Samtaney and Sreenivasan2022a). The mean vertical temperature gradient $\partial T/\partial z$ decreases with increasing $Ra$ in the non-rotating case, whereas it changes in a specific manner in rotating convection (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b; Cheng et al. Reference Cheng, Madonia, Aguirre Guzmán and Kunnen2020; Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022). We compute mean vertical temperature gradient in the bulk region, $\langle \partial T/\partial z \rangle _\mathrm {bulk}$, by performing an average over the bulk volume with $z/H \in [0.25, 0.75]$, and plot it as a function of $Ra$ in figure 11(a) for $Pr = 0.021$. The gradient remains close to unity and does not change significantly near the onset of non-rotating convection (green stars). Thus the bulk flow state in the vicinity of the onset does not differ much from the unmixed conduction state with $\partial T/\partial z = -1$. For $Ra > 10^8$, however, $-\langle \partial T/\partial z \rangle _{bulk}$ decreases monotonically with $Ra$, but even for $Ra = 10^{10}$, low-$Pr$ RBC in the slender cell possesses a higher gradient than in the well-mixed case of extended domains.

Figure 11. Mean vertical temperature gradient in the bulk region between $z = 0.25H$ and $z = 0.75H$ as a function of $Ra$ from non-rotating (green stars) and rotating (open symbols) slender cells for (a) $Pr = 0.021$ and (b) $Pr = 1$. Mean gradient decreases monotonically with $Ra$ in the non-rotating convection, whereas a non-monotonic trend is observed in the rotating convection. Solid and dashed curves are guides to the eye and not the best fits. The dash-dotted vertical line in (b) indicates the transition $Ra \approx 23\,Ek^{-4/3}$ between the cellular and plumes regimes, as found by Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014). Dashed vertical lines in both plots correspond to $Ro = 0.2$. Non-rotating data in (b) are taken from Iyer et al. (Reference Iyer, Scheel, Schumacher and Sreenivasan2020).

The variation of $\langle \partial T/\partial z \rangle _{bulk}$ in figure 11(a) for rotating convection (red circles) is different. It has been known from simulations of the asymptotically reduced equations (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b) as well as DNS of RRBC (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022) that the temperature gradient decreases steeply with $Ra$ in the cellular and columnar regimes, which occur in the vicinity of convective onset for moderate and large Prandtl numbers. With further increase of $Ra$, the gradient increases in the plumes regime and nearly saturates in the geostrophic regime, where the vertical coherence is lost (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014). For higher Rayleigh numbers in the rotation-affected regime, the gradient decreases again (Cheng et al. Reference Cheng, Madonia, Aguirre Guzmán and Kunnen2020). Figure 11(a) shows that the gradient for $Ek = 1.45 \times 10^{-6}$ decreases rapidly near the onset and starts to increase at $Ra = 2 \times 10^8$, before decreasing again for $Ra \geq 5 \times 10^8$.

Figure 11(b) shows the temperature gradient for $Pr = 1$ from the rotating case ($Ek = 10^{-6}$) and the non-rotating case (data taken from Iyer et al. Reference Iyer, Scheel, Schumacher and Sreenivasan2020), both for slender cells. The bulk temperature gradient varies with $Ra$ qualitatively the same way as in the low-$Pr$ rotating convection. Near the critical Rayleigh number, the gradient follows an $Ra^{-0.96}$ scaling indicated by the solid blue curve. This scaling is consistent with the onset results in simulations of asymptotic equations (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b) as well as with the DNS (King et al. Reference King, Stellmach and Buffett2013; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022) for moderate Prandtl numbers. With increasing $Ra$, the gradient decreases more slowly before increasing from $Ra = 2 \times 10^9$ up to $Ra \approx 6 \times 10^9$. As discussed earlier, an increasing gradient with $Ra$ is a characteristic of the plumes region. Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) performed DNS in a horizontally periodic domain with both no-slip and free-slip plates, and observed that the transition for $Pr = 1$ from the cellular to plumes region occurs at $Ra \approx 23\,Ek^{-4/3}$ for the no-slip case. This corresponds to $Ra \approx 2.3 \times 10^9$ for the slender data; we indicate this transition $Ra$ by the dash-dotted vertical line in figure 11(b). It is interesting that the transition $Ra$ found for a horizontally periodic domain identifies the transition for the slender data quite well. It is observed in experiments (Cheng et al. Reference Cheng, Madonia, Aguirre Guzmán and Kunnen2020) and DNS (Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022) that the temperature gradient decreases as $Ra^{-0.21}$ in the rotationally affected regime, when the thermal forcing is significantly stronger than the critical value for the onset. The slender data at the largest Rayleigh numbers in figure 11(b) nearly follow this scaling. Thus the temperature gradient with $Ra$ in the rotating slender cell is qualitatively similar to that in wider cells, indicating again the dominance of rotation over confinement.

To see if the data in figure 11(a) exhibit the scaling features just mentioned for moderate Prandtl numbers, we indicate the $Ra^{-0.96}$ scaling by a solid red curve, but find that the gradient near the onset decreases more slowly; instead, the data follow a $\langle \partial T/\partial z \rangle _{bulk} \sim Ra^{-0.21}$ scaling (red dash-dotted curve). This is possibly an indication of a different flow state near the onset in low-$Pr$ convection. The $Ra^{-0.21}$ scaling in the rotation-affected regime is also indicated as a red dashed curve; we find from this exercise that the gradient for the largest $Ra$ in the low-$Pr$ case is not very different. The dashed vertical lines in figure 11 correspond to $Ro = 0.2$, which suggests that the $Ra^{-0.21}$ scaling occurs when $Ro > 0.2$ and the rotational constraint in bulk region relaxes gradually.

The thickness of the viscous boundary layer (VBL) near the horizontal plates decreases with increasing $Ra$ in non-rotating convection (King et al. Reference King, Stellmach and Buffett2013; Scheel & Schumacher Reference Scheel and Schumacher2017; Bhattacharya et al. Reference Bhattacharya, Pandey, Kumar and Verma2018). In rotating convection, however, the VBL – also known as the Ekman layer – is controlled by the Ekman number; for weak thermal forcings, i.e. in a rotationally controlled regime, the Ekman layer thickness scales as $\sqrt {Ek}$ (King et al. Reference King, Stellmach and Buffett2013; Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022). The VBL thickness $\delta _u$ is frequently determined using the r.m.s. horizontal velocity profile $u_h (z)$, where $u_h = \sqrt {\langle u_x^2 + u_y^2 \rangle _{A,t}}$. Due to the imposed no-slip condition in our simulations, $u_h$ vanishes at the plates and increases rapidly as one moves away from them. We estimate $\delta _u$ as the distance of the first local maximum in the $u_h(z)$ profile from the horizontal plate. We compute $\delta _u$ at both the top and bottom plates, and show the averaged thickness as a function of $Ra$ in figure 12(a) for both the non-rotating and rotating slender cells. For the non-rotating case (green stars), $\delta _u$ decreases with $Ra$. In figure 12(b), we plot the same data as a function of $Re$, which suggests that data for $Re > 10^3$ may be described by a single power law. The best fit for this regime yields a $\delta _u = 0.05\,Re^{-0.26}$ scaling, which is in qualitative agreement with the $Re^{-1/4}$ scaling observed in wider convection domains for moderate and high Prandtl numbers (King et al. Reference King, Stellmach and Buffett2013).

Figure 12. (a) Viscous boundary layer thickness for $Pr = 0.021$, averaged over both the horizontal plates, decreases with $Ra$ in the non-rotating slender cell, whereas it remains constant at low $Ra$ in the rotating cell. (b) Thickness in the non-rotating cell as a function of $Re$. The best fit for $Re > 10^3$ shows that $\delta _u \sim Re^{-1/4}$. (c) Normalized Ekman layer thickness $\delta _u/\sqrt {Ek}$ remains a constant for a wider range of $Ra$ for $Pr = 1$ than for $Pr = 0.021$ simulations. (d) The horizontal velocity profile in the rotating slender cell (solid curves) for $Pr = 0.021$ follows the analytical Ekman layer profile (dashed curves) perfectly up to $Ra = 10^8$, but deviates for larger Rayleigh numbers.

The Ekman layer thickness for $Ek = 1.45 \times 10^{-6}$ in figure 12(a) is nearly independent of $Ra$ for $Ra \leq 10^8$. The constancy of $\delta _u$ suggests that the VBL in this regime behaves as the classical Ekman layer, which results from the balance between the viscous and Coriolis forces (King et al. Reference King, Stellmach and Buffett2013). Figure 12(a) also reveals that a considerable variation in $\delta _u$ is observed for higher Rayleigh numbers. Further, the difference between the rotating and non-rotating data becomes very small for $Ra \geq 10^9$, which indicates the increasing dominance of thermal forcing over rotation as $Ra$ increases. To see the $Ek$ dependence of $\delta _u$, we plot the normalized thickness $\delta _u/\sqrt {Ek}$ as a function of $Ra$ in figure 12(c), and also include the data from the $Pr = 1$, $Ek = 10^{-6}$ simulations. The figure shows that the $\delta _u \sim \sqrt {Ek}$ scaling is indeed observed for both the Prandtl numbers at low thermal forcings and the prefactor $\approx 3$ for $Pr = 0.021$, and $\approx 3.5$ for $Pr = 1$ simulations. These prefactors are in the range of values reported in RRBC in wider domains (King et al. Reference King, Stellmach and Buffett2013; Aguirre Guzmán et al. Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022). Figure 12(c) also shows that the range of Rayleigh numbers over which the VBL is of the Ekman type is wider for $Pr = 1$ than for $Pr = 0.021$, which indicates the inertial nature of low-$Pr$ RRBC and is consistent with the findings of Aguirre Guzmán et al. (Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022).

We further probe the Ekman layer in the slender cell by investigating the form of the r.m.s. horizontal velocity profile $u_h(z)$ near the plates. For the classical Ekman layer above a no-slip plate, the velocity profile can be obtained analytically by considering a geostrophic bulk flow, where the horizontal pressure gradients are balanced by the Coriolis forces, and assuming that the same horizontal pressure gradients exist within the boundary layer region. Following Kundu & Cohen (Reference Kundu and Cohen2004) and Aguirre Guzmán et al. (Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022), we find that $u_h(z)$ near the plate can be described by

(6.1)\begin{equation} u_h(z) = U_h [1 - 2 \cos(z/\delta_U) \exp({-z/\delta_U}) + \exp({-2z/\delta_U})]^{1/2}. \end{equation}

Here, $U_h = \sqrt {U_x^2+U_y^2}$ is the r.m.s. horizontal velocity in the geostrophic bulk, with $U_x$ and $U_y$ being the horizontal velocity components. The parameter $\delta _U$ corresponds to the thickness of the Ekman layer. In figure 12(d), we show $u_h(z)$ for four Rayleigh numbers from the simulations at $Pr = 0.021$, $Ek = 1.45 \times 10^{-6}$ as solid curves. We fit these profiles using (6.1), and determine the parameters $U_h$ and $\delta _U$, and the resulting profiles obtained from (6.1) with the fitted parameters are exhibited as dashed curves in figure 12(d). We observe that the profiles for $Ra \leq 10^8$ can be described excellently by the analytical profile (6.1). However, deviation starts to appear for $Ra \geq 1.5 \times 10^8$. Figure 12(d) exhibits that (6.1) still describes the near-wall profiles for all Rayleigh numbers. Thus the VBL in the slender cell for $Pr = 0.021$ is of the Ekman type only up to $Ra = 10^8$, which is consistent with the inference from figure 12(c). Note that similar results were reported in RRBC in horizontally periodic boxes by Aguirre Guzmán et al. (Reference Aguirre Guzmán, Madonia, Cheng, Ostilla-Mónico, Clercx and Kunnen2022).