3.1 Onset of convection

In this section, we focus on the flow near the onset of convection. The combined effect of buoyancy and rotation generates a flow structured in columns at the onset. This is illustrated by the contours of vertical velocity obtained for different values of $E$ in figure 4. Note that the columns are exclusively confined within the TC. This can be inferred from the sharp drop of vertical velocity visible all along the TC boundary. It can also be seen from the radial profiles of velocity in figure 12(a,c). It is noteworthy that the configuration of our experiment favours confinement within the TC because (i) the heat flux is generated along the whole horizontal section of the TC and nowhere outside it and (ii) gravity is always parallel to the TC boundary, unlike in the Earth where they intersect. Hence, the buoyancy force never directly acts as to break the TP constraint in LEE, unlike in the Earth. However, the small residual flow outside the TC is further attenuated by an increase in background rotation (see figure 12 a–d). Conversely, qualitative experiments conducted without rotation show a very different picture where plumes take the entire space above the heater, for the flow to return entirely outside of the would-be TC. Considering the sharpness of the velocity jump across the TC boundary, these elements give a good evidence that the confinement we observe mainly results from TP constraint. Columns become thinner at lower values of $E$ , a tendency that has been previously observed both in spherical and plane geometries (Chandrasekhar Reference Chandrasekhar1961; Sreenivasan & Jones Reference Sreenivasan and Jones2006; Aujogue, Pothérat & Sreenivasan Reference Aujogue, Pothérat and Sreenivasan2015).

In figure 3 we show that variations of the critical Rayleigh number with $E$ follow a scaling of $Ra_{c}=(32.3\pm 4)\times E^{-1.29\pm 0.05}$ . Despite a very different geometry, this scaling is in good agreement with the theoretical prediction of $Ra_{c}=22.3\times E^{-4/3}$ for rotating convection in an infinitely extended plane layer (Chandrasekhar Reference Chandrasekhar1961). This result suggests that the critical Rayleigh number is not measurably affected by the geometry differences between a plane and our TC and reflects a very robust feature of plane rotating convection.

Figure 3.

Critical Rayleigh number $Ra_{c}$ and critical wavenumber $a_{c}=2\unicode[STIX]{x03C0}R/r_{0}$ at onset versus the Ekman number $E$ for three geometries: present case (symbols and dotted lines), infinite plane layer (solid lines) and wall modes in a finite cylinder obtained from the theory of Zhang & Liao (Reference Zhang and Liao2009) ((3.3) and (3.4)), dashed lines). Raw and interpolation data for this figure are available as the supplementary material at https://doi.org/10.1017/jfm.2018.77.

Figure 4.

Average over time of the vertical component of velocity from PIV measurement above the liquid heater at the onset of convection. The averaging time is 15 times smaller than the precession time scale $\unicode[STIX]{x1D70F}_{p}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}_{p}$ . Velocities are normalised by the maximum velocity of the flow. Here $z$ and $r$ are normalised by the maximum height above the heater and the radius of the heater respectively. The blue line represents the inner boundary of the glass dome. The red arrows point to the position of the TC. (a)  $E=2.51\times 10^{-5}$ and $Ra=3.48\times 10^{7}$ . (b)  $E=3.36\times 10^{-6}$ and $Ra=3.82\times 10^{8}$ .

We shall now analyse the horizontal size of the convective structures present at the onset. Since translational invariance in the horizontal plane is lost in our geometry, we extract the horizontal size of convective structures by seeking the separation $r_{0}$ corresponding to the first zero of the spatial correlation function built from $u_{r}$ , and averaged over time and $z$ :

(3.1) $$\begin{eqnarray}\displaystyle C_{u_{r}}(\unicode[STIX]{x1D6FF}r)=\left\langle \int _{{\mathcal{V}}}u_{r}(r+\unicode[STIX]{x1D6FF}r)u_{r}(r)\,\text{d}r\,\text{d}z\right\rangle _{t}, & & \displaystyle\end{eqnarray}$$

where ${\mathcal{V}}$ represents the intersection of the meridional plane lit by the PIV laser and the region inside the TC. At the onset, the associated wavenumber $a_{c}=2\unicode[STIX]{x03C0}R/r_{0}$ can be compared with the critical wavenumber predicted for the onset of rotating convection in an infinite plane layer (figure 3). In the TC geometry, we find a scaling of

(3.2) $$\begin{eqnarray}\displaystyle a_{c}=(0.58\pm 0.08)\times E^{-0.32\pm 0.05} & & \displaystyle\end{eqnarray}$$

when the plane layer theory predicts $a_{c}=1.65\times E^{-1/3}$ (Aujogue et al. Reference Aujogue, Pothérat and Sreenivasan2015). Although both structure sizes exhibit the same scaling exponent, critical wavenumbers are significantly lower in the TC geometry than for the infinite plane layer. We shall see in § 3.2 that the reason for this discrepancy can be linked to the topological structure of the critical modes.

Lastly, a remarkable feature of the onset of rotating convection in plane layers is that for the values of $Pr$ considered in this paper, linear stability predicts a steady critical mode (Clune & Knoblauch Reference Clune and Knoblauch1993). By contrast, in all our measurements, we found a time-dependent flow at the onset. Inspection of the flow in horizontal planes reveals that the convective plumes are subject to a slow retrograde precession. The variations of the corresponding frequency $\unicode[STIX]{x1D714}_{p}$ derived from the maximum velocity along $\unicode[STIX]{x1D703}$ in the horizontal plane and normalised by the background angular frequency are reported in figure 7. The precession at the onset of convection and beyond has been studied theoretically and experimentally in Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993) and Zhong et al. (Reference Zhong, Ecke and Steinberg1993) in rigid rotating cylinders of various aspect ratios. In these studies, the authors showed that the loss of translational symmetry in the radial direction necessarily induced a precession in the critical mode and that the corresponding frequency normalised by the background rotation $\unicode[STIX]{x1D6FA}$ , $\unicode[STIX]{x1D714}_{p_{0}}$ obeyed a scaling of the form $\unicode[STIX]{x1D714}_{p_{0}}=\unicode[STIX]{x1D6FF}E^{-1}$ . Experiments by Ecke, Zhong & Knobloch (Reference Ecke, Zhong and Knobloch1992) determined a value of $\unicode[STIX]{x1D6FF}=0.1$ for a radius-to-height aspect ratio of $\unicode[STIX]{x1D6E4}=1$ . Our measurements produce a value of $\unicode[STIX]{x1D6FF}=0.07\pm 0.005$ . This value is close to that found by Ecke et al. (Reference Ecke, Zhong and Knobloch1992), despite a lower aspect ratio (measured at the centre of the heater) of $\unicode[STIX]{x1D6E4}=R/d=5/12$ .

The precession of convective structures at the onset in a cylindrical geometries is normally associated with the onset of modes that are localised either at the centre of the cylinder or near the wall, respectively centre modes and wall modes (Goldstein et al. Reference Goldstein, Knobloch, Mercader and Net1993). The values of the precessing frequencies and of $\unicode[STIX]{x1D6FF}$ we find point to wall modes rather than centre modes (Ecke et al. Reference Ecke, Zhong and Knobloch1992). Nevertheless, in cylinders bounded by solid walls, the onset of wall modes normally takes places at a lower Rayleigh number (denoted $Ra_{W}$ ) than the steady modes that ignite plane convection for the same value of $E$ . Zhang & Liao (Reference Zhang and Liao2009) provide an estimate for $Ra_{W}$ and the corresponding azimuthal wavenumber $a_{W}$ , which are reported in figure 3 and expressed in our notations and for the present configuration as

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle Ra_{W}=31.81E^{-1}+46.49E^{-2/3}, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle a_{W}=2.118-12.15E^{1/3}. & \displaystyle\end{eqnarray}$$

The fact that in the TC, $Ra_{c}$ follows the scaling for plane convection rather than that of wall modes is perhaps explained by the fact that unlike a solid wall, the Taylor–Proudman constraint is absent when the fluid is at rest. Wall modes would be expected to occur at a lower critical Rayleigh number than the onset of plane convection. Since, however, no confinement exists at such Rayleigh numbers, the mechanism for their onset is absent. On the other hand, as soon as convection starts, i.e. for Rayleigh numbers close to those for which the plane layer would be unstable, the TP constraint becomes active and selects modes that are closer to those observed in a cylinder than those observed between infinite planes. Hence, the TP constraint may not influence the flow right at the onset, but only once unstable structure have sufficiently developed.

In summary, the study of critical Rayleigh, plume size and precession frequency shows that the onset of convection in a TC corresponds to a hybrid behaviour between those of convection in an infinite plane layer and in a rotating cylinder. Certainly, the scaling for the critical Rayleigh number from the plane layer theory is reproduced in the TC but size and time dependence of the flow are to some extent more accurately described by the phenomenology of convection in rigid rotating cylinders.

3.2 Supercritical flow patterns

We shall now explore the evolution of the patterns from the onset of convection into the supercritical regime, focusing on two values of the Ekman number $E=1.15\times 10^{-5}$ and $E=6.36\times 10^{-6}$ , for critical parameters $R_{c}=(Ra-Ra_{c})/Ra_{c}$ in the range 0.1–12.

Figure 5(a–f) illustrates the development of convective patterns for a criticality in the range $0.1\leqslant R_{c}<12$ and $E=1.15\times 10^{-5}$ . Near the onset of convection ( $R_{c}=0.13$ , figure 5 a) we observe a behaviour found in the linear stability analysis performed by Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993) on a rapidly rotating cylinder with no-slip boundary conditions at the top and bottom and adiabatic boundary conditions on the side walls. In this study, the authors characterised two different types of convective mode at onset, labelled fast and slow. The fast modes correspond to motion near the lateral wall of the cylinder. The slow modes describe convective patterns localised at the centre of the cylinder. Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993) also showed that the changeover between fast and slow modes is strongly dependent on the aspect ratio of the cylinder. In figure 5(a), we observe a structure similar to a fast mode of azimuthal wavenumber $m=2$ . For this wavenumber, Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993) predicted that the fast mode was the most unstable below an aspect ratio of 1.84, a condition satisfied in the TC geometry of our experiment. Note, however that the fast/wall mode structure is not as sharp as for higher levels of criticality. This is in part due to the technical difficulty in accurately resolving the small velocity differences near the onset. It may also reflect that the TC boundary exerts a somewhat weaker influence near the onset than the solid walls of the cylinder of Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993).

Figure 5.

Time-averaged velocity fields (arrows) with vorticity field (colour bar) in the horizontal plane at $z/d=3/4$ for different Rayleigh numbers with $E=1.15\times 10^{-5}$ . The blue line represents the boundary of the glass dome. The red line represents the position of the heater that defines the TC. Wall modes can be identified where vorticity extrema are present near the boundary of the TC (e.g. a,c). The averaging time is 15 times smaller than the precession time scale $\unicode[STIX]{x1D70F}_{p}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}_{p}$ . (a) $R_{c}=0.13$ , fast wall mode (WM) at onset. (b) $R_{c}=1.26$ , quasi-geostrophic (QG) mode. (c) $R_{c}=3.58$ , QG and WM interaction. (d) $R_{c}=6.29$ , merging of centre modes. (e) $R_{c}=8.25$ , merging of centre modes. (f) $R_{c}=11.27$ , central retrograde vortex.

Figure 6.

Time-averaged velocity fields (arrows) with vorticity field (colour bar) in the horizontal plane at $z/d=3/4$ for different Rayleigh numbers with $E=6.36\times 10^{-6}$ . The blue line represents the boundary of the glass dome. The red line represents the position of the heater that defines the TC. Wall modes can be identified where vorticity extrema are present near the boundary of the TC (e.g. c). The averaging time is 15 times smaller than the precession time scale $\unicode[STIX]{x1D70F}_{p}=2\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}_{p}$ . (a) $R_{c}=0.35$ , localised structure near the centre at onset. (b) $R_{c}=1.26$ , modulated modes. (c) $R_{c}=2.32$ , interaction between wall modes and centre modes. (d) $R_{c}=4.40$ , merging of centre modes. (e) $R_{c}=8.17$ , merging of centre modes. (f) $R_{c}=11.91$ , central retrograde vortex.

Figure 7.

Evolution of the precession $\unicode[STIX]{x1D714}_{p}$ and the wavenumber $a$ as function of the degree of criticality. Raw and interpolation data for this figure are available as the supplementary material.

For $R_{c}=1.26$ (figure 5 b), several columns are gathered around the centre of the TC. These are reminiscent of structures which Aurnou et al. (Reference Aurnou, Andreadis, Zhu and Olson2003) call quasi-geostrophic modes, with the difference that in their experiment, dye visualisations suggested that they were forming on the outside of the TC. When further increasing $R_{c}$ ( $R_{c}=4.58$ ; figure 5 c), we see an interaction between these centre modes and ‘wall modes’ forming near the side boundary. In figure 5(d,e), we observe that for $R_{c}\geqslant 6.29$ the centre modes are merging into one larger structure, which evolves into a large central, retrograde vortex in the last stage of the convection observed in the experiment (figure 5 f, $R_{c}=11.27$ ). The evolution towards a central retrograde vortex for critical parameters exceeding $R_{c}=10$ was also found by Zhong et al. (Reference Zhong, Ecke and Steinberg1993) in experiments on convection in a rotating cylinder of aspect ratio $\unicode[STIX]{x1D6E4}=1$ (i.e. wider than in the present case).

Figure 6(a–f) shows the flow at a smaller Ekman number, $E=6.36\times 10^{-6}$ , for $R_{c}\in [0.35\times Ra_{c},11.91\times Ra_{c}]$ . Near onset ( $R_{c}=0.35$ , figure 6 a), the convective patterns are much smaller than for a comparable criticality at $E=1.15\times 10^{-5}$ . When the criticality increases, we observe modes that correspond to the modulated modes described by Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993). These feature spiralling arms (figure 6 b) and were also observed experimentally by Zhong et al. (Reference Zhong, Ecke and Steinberg1993) for a comparable level of criticality ( $R_{c}=2.56$ ). Zhong et al. (Reference Zhong, Ecke and Steinberg1993) showed that these spiralling patterns arise from the outer wall of the cylinder as azimuthal mode. In the TC, we observe such an azimuthal mode with a corresponding wavenumber $m=2$ . Such large differences between supercritical patterns at different Ekman numbers were theoretically predicted by Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993), whose analysis shows that when $E$ is varied, the lowest critical Rayleigh number is alternatively achieved by either a fast or a slow precessing mode. At higher criticality, convective patterns reflect a combination of wall modes and centre modes (at $R_{c}=2.32,4.40,9.17$ in figure 6(c–e)), as for $E=1.15\times 10^{-5}$ . Similarly to the more slowly rotating case, convective patterns converge towards a central retrograde structure at the highest levels of supercriticality explored here (figure 6 f).

For both values of $E$ , the main features of the flow patterns (alternative presence of fast and slow modes near the onset of convection depending on $E$ , evolution towards a large retrograde vortex for $R_{c}\gtrsim 10$ ) support the view that the convection in the TC behaves as convection in a solid cylinder rather than in an infinite plane layer. It is also interesting to note that the flow patterns discovered by Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993) and Zhong et al. (Reference Zhong, Ecke and Steinberg1993) at relatively high values of $E$ ( ${\sim}10^{-3}{-}10^{-2}$ ) remain dominant at the much lower values explored in our experiment ( ${\sim}10^{-6}{-}10^{-5}$ ). Such robustness may indicate that convection is in an asymptotic regime of rotation as far as flow structures are concerned and that similar structures might also be found in regimes of even more rapid rotation, such as the Earth’s.

It should also be stated that other authors mention the existence of instabilities at the rim of the TC (see Cui & Street Reference Cui and Street2001; Aurnou et al. Reference Aurnou, Andreadis, Zhu and Olson2003). While it is difficult to distinguish such instabilities from convective instabilities within the cylinder from visualisations only, Maxworthy & Narimousa (Reference Maxworthy and Narimousa1994) found that such instabilities occur for a Rossby number based on vertical motion $Ro^{\ast }=u_{z}/2\unicode[STIX]{x1D6FA}$ greater than 0.28. This parameter remains one or two orders of magnitude below this value in most of our experiments, and below 0.1 in the most supercritical cases (for $R_{c}\simeq 10$ ). This is a good indication that the structures we observe are convective patterns, rather than rim instabilities.

3.3 Evolution of the plume size and precession frequencies in supercritical regimes

The succession of supercritical patterns is reflected in the evolution of the dominating wavelengths and precession frequencies with criticality $R_{c}$ gathered in figure 7. We find that beyond the onset of convection the dominating wavenumber follows a scaling of the form

(3.5) $$\begin{eqnarray}\displaystyle aE^{1/3}=(0.5\pm 0.025)\times (R_{c}+1)^{-0.45\pm 0.05}. & & \displaystyle\end{eqnarray}$$

This scaling extends the scaling for $a_{c}(E)$ found at the onset of convection to supercritical regimes. It implies that as convection becomes more intense, the flow rearranges itself with fewer larger structures that are more efficient to carry the heat flux across the TC up to the point where only one structure is left for $R_{c}\gtrsim 10$ .

The precession frequency first sharply increases in the weakly supercritical regime and subsequently saturates. It is nevertheless difficult to tell whether an asymptotic value is reached in the limit $R_{c}\rightarrow \infty$ . Using a Landau model for the bifurcation, Goldstein et al. (Reference Goldstein, Knobloch, Mercader and Net1993) showed that in weakly supercritical regime, the precession normalised by $\unicode[STIX]{x1D6FA}$ should vary as:

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{p}=E^{-1}(\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D719}R_{c})+O(R_{c}^{2}), & & \displaystyle\end{eqnarray}$$

where the values of constants $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D719}$ depend on the aspect ratio of the cylinder (with $\unicode[STIX]{x1D6FF}\rightarrow 0$ in the limit of large aspect ratio to recover the stationary onset of convection in an infinitely extended plane layer, see Chandrasekhar (Reference Chandrasekhar1961)). The values of $\unicode[STIX]{x1D6FF}=0.1$ and $\unicode[STIX]{x1D719}=5$ were obtained experimentally by Ecke et al. (Reference Ecke, Zhong and Knobloch1992) and Zhong et al. (Reference Zhong, Ecke and Steinberg1993) for a cylinder of aspect ratio $\unicode[STIX]{x1D6E4}=1$ and values of $E$ greater than $10^{-3}$ . The variations of $\unicode[STIX]{x1D714}_{p}(R_{c})$ extracted from experiments at several Ekman numbers for $R_{c}>0$ are reported in figure 7. They are best fitted over the widest measured range of values by an exponential law of the form:

(3.7) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{p}=(3\pm 0.5)\times E^{-1}(1-\text{e}^{(-0.24\pm 0.01)R_{c}})+0.1\times E^{-1}. & & \displaystyle\end{eqnarray}$$

Expanding (3.7) to $O(R_{c})$ and identifying with (3.6), we find that $\unicode[STIX]{x1D719}=0.534\pm 0.105$ . The value of $\unicode[STIX]{x1D6FF}$ is consistent with the findings of Ecke et al. (Reference Ecke, Zhong and Knobloch1992), but the precession frequency appears to vary significantly less in the supercritical regime in the present case than in the case of a solid rotating cylinder. It is, however, difficult to attribute this difference to any of the factors that differ between the two problems: the difference in shape of the upper domain boundary, a different aspect ratio, three orders of magnitude difference in the Ekman number, different mechanical and thermal boundary conditions at the lateral boundary of the cylinder. Nevertheless, the precessing motion at onset and in supercritical regimes adds further support to the view that the phenomenology of convection in the tangent cylinder obeys some of the mechanisms of convection as in a solid rotating cylinder, both at the onset and in supercritical regimes.

4.1 Heat flux through the tangent cylinder

Figure 8.

Variations of the Nusselt number $Nu$ emphasising the scaling $Nu/E^{2}\simeq 0.2Ra^{0.33}$ for large Rayleigh numbers (a) and the scaling $Nu\simeq 0.38E^{2}Ra^{1.58}$ at moderate Rayleigh numbers (b). Raw and interpolation data for this figure are available as the supplementary material.

Figure 9.

Variations of $Nu^{\ast }$ with $Ra_{q}^{\ast }$ . Raw and interpolation data for this figure are available as the supplementary material.

The efficiency of convection is best measured by its ability to transport heat through a fluid layer. Here we characterise the variations of the heat flux with $Ra$ and $E$ by means of the Nusselt number which represents the ratio of the heat flux in the fluid to the purely conductive heat flux:

(4.1) $$\begin{eqnarray}\displaystyle Nu=\frac{Fd}{k\unicode[STIX]{x0394}T}, & & \displaystyle\end{eqnarray}$$

where we recall that $F$ is the heat flux through the top surface of the heater, obtained from the difference between the temperature of the heat-carrying fluid at the inlet and outlet of the heater and $d$ is the height of fluid above the heater. The variations of $Nu$ with $Ra$ are reported in figure 8 for several values of $E$ . Since $F$ is measured at the heater, it reflects the total heat transferred across the entire dome. As all of the flux transits through the TC, $F$ is an upper bound for the total heat flux through the upper surface of the TC because thermal losses occur at the lateral boundary of the heater. Given the low values of the velocity near that boundary, these losses are most likely predominantly of conductive nature, especially at low criticality. Two regions of parameters clearly appear. In the large $Ra$ limit, the heat transfer becomes independent of the rotation and follows a scaling of the form (see a):

(4.2) $$\begin{eqnarray}\displaystyle Nu=(0.2\pm 0.04)Ra^{0.33\pm 0.03}. & & \displaystyle\end{eqnarray}$$

This suggests that in these strongly supercritical regimes, heat transfer is not influenced by the Coriolis force anymore. Indeed, in the range of Rayleigh and Prandtl numbers we considered, similarly low exponents of $Ra$ in the range $1/4$ – $3/7$ are expected for plane non-rotating Rayleigh–Bénard convection (Grossmann & Lohse Reference Grossmann and Lohse2000) and in spherical shells with radial gravity (Gastine, Wicht & Aurnou Reference Gastine, Wicht and Aurnou2015). At moderate Rayleigh numbers, by contrast, ( $Ra\gtrsim 8\times 10^{7}$ for $E=1.15\times 10^{-5}$ , $Ra\gtrsim 1\times 10^{8}$ for $E=6.35\times 10^{-6}$ and $Ra\gtrsim 1.5\times 10^{8}$ for $E=4.46\times 10^{-6}$ , see b), $Nu$ exhibits a strong dependence on $E$ as well as $Ra$ , of the form:

(4.3) $$\begin{eqnarray}\displaystyle Nu=(0.38\pm 0.02)\times Ra^{1.58\pm 0.06}E^{2\pm 0.04}. & & \displaystyle\end{eqnarray}$$

This law is very close to the theoretical scaling of $Nu\sim 0.15Ra^{3/2}E^{2}$ put forward by Julien et al. (Reference Julien, Knobloch, Rubio and Vasil2012) (see also Gastine et al. Reference Gastine, Wicht and Aubert2016) as a signature of the diffusivity-free regime. These authors found that this scaling was verified in rotating convection in a spherical shell with radial gravity for a range of values of $Ra$ and $E$ that is essentially the same as its range of validity in LEE. Their theoretical scaling argument, however, does not invoke geometry nor the orientation of the buoyancy force. It relies on the assumptions that (i) the largest contribution of the temperature gradient originates in the bulk and not in boundary layers, (ii) in the limit of no rotation, the ultimate regime of classical Rayleigh–Bénard convection is recovered (Kraichnan Reference Kraichnan1962), (iii) in the limit where rotation dominates inertia, the heat flux solely depends on $R_{c}$ (King, Stellmach & Aurnou Reference King, Stellmach and Aurnou2012). Hence, this scaling can be reasonably interpreted as the signature of the diffusivity-free regime of rotationally dominant convection in the TC configuration too. Experiments and numerical simulations on convection in a rotating cylinder filled with water (King et al. Reference King, Stellmach and Aurnou2012; Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015) also exhibit a similar transition between steep variations of $Nu(Ra)$ in the weakly supercritical regime and a more a strongly supercritical regime where $Nu\sim Ra^{0.32}$ that is very close to (4.2). In this work, however, the scaling law at low $Ra$ , was found to strongly depend on $E$ , with an exponent varying from $Nu\sim (RaE^{4/3})^{6/5}$ at low Ekman numbers to $Nu\sim (RaE^{4/3})^{3.6}$ (for $E=10^{-7}$ ).

An alternative but equivalent way to analyse the variations of the heat flux consists of seeking how the flux normalised by rotation varies with the Rayleigh number based on the heat flux itself. One advantage of this approach is to quantify convection in terms of the available buoyancy. Buoyancy may indeed result from a heat flux at the boundary but also from a solute mass flux, as in the core of the Earth and other planets. On this basis, we follow Aubert et al. (Reference Aubert, Brito, Nataf, Cardin and Masson2001), Aubert (Reference Aubert2005), Christensen & Aubert (Reference Christensen and Aubert2006) and introduce the modified Nusselt number, Rayleigh number and a modified diffusionless Rayleigh number based on the heat flux, respectively defined as

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle Nu^{\ast }=Nu\times E\times Pr^{-1}, & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle Ra^{\ast }=Ra\times E^{2}\times Pr^{-1}, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle Ra_{q}^{\ast }=Ra^{\ast }Nu^{\ast }. & \displaystyle\end{eqnarray}$$

Cheng & Aurnou (Reference Cheng and Aurnou2016) stress that scalings of the form $Nu\sim Ra^{\unicode[STIX]{x1D6FC}}$ and $Nu^{\ast }\sim {Ra_{q}^{\ast }}^{\unicode[STIX]{x1D6FD}}$ satisfy $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}/(1+\unicode[STIX]{x1D6FC})$ . Here $\unicode[STIX]{x1D6FC}=0.33\pm 0.03$ , $\unicode[STIX]{x1D6FC}/(1+\unicode[STIX]{x1D6FC})\in [0.23,0.27]$ and $\unicode[STIX]{x1D6FD}=0.3$ . Hence, these scalings offer an alternative way of representing the data from figure 9, rather than new data. Christensen (Reference Christensen2002) found that these quantities obeyed a scaling law of the form $Nu^{\ast }\sim (Ra_{q}^{\ast })^{5/9}$ . This result was obtained with numerical simulations in a spherical shell geometry for $Ra_{q}^{\ast }\in [10^{-7},10^{-3}]$ . Aurnou (Reference Aurnou2007) identified this scaling as indicative of a rapidly rotating regime. Further, using the data of Sumita & Olson (Reference Sumita and Olson2003), it was shown that for low values of $Ra_{q}^{\ast }$ the relation between $Nu^{\ast }$ and $Ra_{q}^{\ast }$ was better fitted with a power law of the form $Nu^{\ast }\sim (Ra_{q}^{\ast })^{0.29}$ . In the present configuration, $Ra_{q}^{\ast }$ varies between $10^{-10}$ and $10^{-6}$ and therefore falls within a similar range to the experiments of Sumita & Olson (Reference Sumita and Olson2003). Collapsed data reported in figure 9 show that points in the regime following the $Nu\sim Ra^{0.33}$ law obey a scaling close to that found by these authors:

(4.7) $$\begin{eqnarray}\displaystyle Nu^{\ast }=(0.0046\pm 0.0005)\times (Ra_{q}^{\ast 1\pm 0.03})^{0.26\pm 0.04}. & & \displaystyle\end{eqnarray}$$

The exponent being closer to 0.29 than $5/9$ confirms that this regime is one where the flow is outside the quasi-geostrophic regime, where convection-driven inertia plays an important role.

4.2 Thermal wind

In axisymmetric geometry, the balance between buoyancy and Coriolis forces gives rise to azimuthal motion, seen through the balance between the curl of these forces:

(4.8) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}u_{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}z}\sim -\frac{g\unicode[STIX]{x1D6FD}}{2\unicode[STIX]{x1D6FA}}\frac{\unicode[STIX]{x2202}T^{\prime }}{\unicode[STIX]{x2202}r}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6FD}$ is the coefficient of thermal expansion and $T^{\prime }$ is the temperature perturbation. The strongest retrograde motion is found at the higher latitude, such as those of our PIV planes represented in figures 5 and 6, located at a latitude of $51.5^{\circ }$ (or, equivalently at $z/d=0.75$ above the heater). On the other hand, the direction of the thermal wind may reverse at low latitude. Hence, we extract from these measurements a radial profile of the azimuthal wind at two different latitudes (see figure 2) by means of an azimuthal and time average of the azimuthal velocity $\langle u_{\unicode[STIX]{x1D703}}(r,z)\rangle _{\unicode[STIX]{x1D703}t}$ : additionally to the high latitude, the second latitude we consider corresponds to a plane close to the heater surface ( $20^{\circ }$ , or, equivalently at $z/d=0.167$ above the heater). Results are plotted in figure 10(a–d).

Figure 10.

Azimuthally and time-averaged radial profiles of azimuthal velocities $\langle u_{\unicode[STIX]{x1D703}}\rangle _{\unicode[STIX]{x1D703}t}(r)$ , for several levels of criticality. (a,b) Data from horizontal plane at $51.5^{\circ }$ . (c,d) Data from horizontal plane at $20^{\circ }$ . (a,c)  $E=1.15\times 10^{-5}$ , (b,d)  $E=6.36\times 10^{-6}$ . Vertical dashed line: TC boundary, vertical solid line: position of the glass dome.

At high latitude (figure 10 a,b), the profiles exhibit a strong negative maximum followed by slightly positive values at larger radii, corresponding to a strong retrograde motion surrounded by a slightly prograde motion. Near the onset of convection, some prograde motion exists near the centre, that disappears as the flow becomes more supercritical. Despite the succession of different patterns observed in the supercritical regime, the intensity of the retrograde motion steadily increases with criticality. For strongly supercritical flows ( $R_{c}\gtrsim 11$ ), velocity patterns show a single structure slightly outgrowing the TC (we shall see in § 5 that motion outside the TC does not necessarily imply that the Taylor–Proudman constraint is broken). This phenomenon appears for slightly higher criticality at higher values of $E$ ( $R_{c}\gtrsim 9$ at $E=1.15\times 10^{-5}$ and $R_{c}\gtrsim 11$ at $E=6.36\times 10^{-6}$ ). In both cases, this confirms that in the most supercritical regimes, retrograde motion progressively invades the high-latitude region of the TC, where the flow becomes dominated by a large central retrograde vortex.

Profiles at low latitude, nearer the solid inner core, provide a better picture of the three-dimensional structure of the thermal wind (figure 10 c,d). In both cases, a prograde wind first develops near the solid core at low levels of criticality and increases in intensity with $R_{c}$ . At $E=1.15\times 10^{-5}$ , the prograde wind starts weakening at the centre from $R_{c}=2.37$ and starts becoming retrograde around $R_{c}=9.92$ . At higher rotation, weakening of the prograde wind near the solid core occurs only for $R_{c}>9$ and we did not reach a regime where it reversed. This difference in levels of criticality for the weakening and reversal of the thermal wind near the solid core can be understood from the scaling for the critical Rayleigh number $Ra_{c}\sim E^{-4/3}$ and (4.8) which imply that $\unicode[STIX]{x2202}_{z}u_{\unicode[STIX]{x1D703}}\propto (R_{c}+1)E^{1/3}$ . From this scaling, thermal wind with a given vertical gradient of $u_{\unicode[STIX]{x1D703}}$ is expected to occur at increasingly high levels of criticality when $E$ is decreased. Hence, the larger velocity gradients corresponding to a thermal wind flowing in opposite directions at high and low latitudes survive at higher levels of criticality when $E$ decreases.

Interestingly, a clear azimuthal prograde jet is present at the edge of the tangent cylinder on the inside. The jet is thinner at faster rotation. It appears for $R_{c}\sim 2.37$ at $E=1.15\times 10^{-5}$ and $R_{c}\sim 4.06$ at $E=6.36\times 10^{-6}$ , and its intensity slightly increases with criticality. One cannot but notice the similarity between this structure and the non-axisymmetric azimuthal jets near the TC inferred by Livermore, Hollerbach & Finlay (Reference Livermore, Hollerbach and Finlay2017) from satellite measurements of time variations of the Earth’s magnetic field. Although the forces driving the jet remain unknown, the authors suggest that they are most likely of magnetic origin, an effect that is absent in the present study. Azimuthal jets were also noticed in measurements in the flow patterns inferred from measurements of the Earth’s magnetic field (Hulot et al. Reference Hulot, Eymin, Langlais, Mandea and Olsen2002).

To conclude the characterisation of the thermal wind, we shall quantify its intensity in terms of the Rossby number $Ro$ , and its variations with the flux-based Rayleigh number, $Ra_{q}^{\ast }$ . The Rossby number, defined as

(4.9) $$\begin{eqnarray}\displaystyle Ro=\frac{U_{m}}{2R\unicode[STIX]{x1D6FA}} & & \displaystyle\end{eqnarray}$$

measures the ratio of inertia to the Coriolis force, where $U_{m}$ is chosen as the maximum retrograde velocity in the averaged profiles of the thermal wind measured in the high-latitude plane (reported in figure 10 a,b). Our results, reported in figure 11, obey the scaling

(4.10) $$\begin{eqnarray}\displaystyle Ro=(5.33\pm 0.3)\times (Ra_{q}^{\ast })^{0.51\pm 0.04}. & & \displaystyle\end{eqnarray}$$

This scaling is very close to the scaling relating the azimuthal velocity scale $U_{\unicode[STIX]{x1D703}}$ and buoyancy $B$ , $U_{\unicode[STIX]{x1D703}}\sim (B/\unicode[STIX]{x1D6FA})^{1/2}$ (here, $Ro\sim {Ra_{q}^{\ast }}^{1/2}$ ) first derived theoretically by Maxworthy & Narimousa (Reference Maxworthy and Narimousa1994) for the thermal wind in oceanic convective plumes. It is derived from the azimuthal curl of a local balance between Coriolis force and buoyancy outside the geostrophic regime, where the excess buoyancy is set by the balance between Coriolis and inertial forces (Maxworthy & Narimousa Reference Maxworthy and Narimousa1994). Both Aurnou et al. (Reference Aurnou, Andreadis, Zhu and Olson2003) and Aubert (Reference Aubert2005) report the same scaling when thermal wind is present respectively in experiments in a TC geometry and in numerical simulations in full spherical shell. Hence (4.10) provides further evidence that the azimuthal motion we are detecting is thermal wind. The vertical gradients of azimuthal velocity induce a departure to geostrophy of order $O(Ro)$ , which is significantly smaller than the vertical velocities associated with the plumes. Hence the thermal wind may not be the main source of the ageostrophy inferred from the variations of heat flux.

Figure 11.

Variations of the Rossby number Scaling $Ro$ with $Ra_{q}^{\ast }$ . Raw and interpolation data for this figure are available as the supplementary material.

Figure 12.

(a,c) $\langle \sqrt{\langle u_{z}(r)^{2}\rangle _{z}}\rangle _{t}$ (top) and $\langle \sqrt{\langle u_{r}(r)_{r}^{2}\rangle _{z}}\rangle _{t}$ (bottom) for several increments of supercriticality and $E=1.15\times 10^{-5}$ . (b,d) Similarly, $\langle \sqrt{\langle u_{z}(r)\rangle _{z}^{2}}\rangle _{t}$ and $\langle \sqrt{\langle u_{r}(r)_{r}^{2}\rangle _{z}}\rangle _{t}$ for several increments of supercriticality and $E=6.36\times 10^{-6}$ .