3.2.1 Nusselt number scaling

As can be seen on figure 1, the weakly nonlinear scaling (3.2) provides an accurate description of the heat transport behaviour up to $Nu\simeq 1.5$ , which approximately corresponds to $Ra\simeq 6\,Ra_{c}$ . When $Nu>2$ , the heat transport increases much more rapidly and enters the nonlinear regime of rotating convection.

Figure 5.

(a) Nusselt number as a function of $\widetilde{Ra}=RaE^{4/3}$ . (b) Corresponding compensated $Nu$ scaling. In both panels, the solid black line corresponds to the scaling $Nu=0.15\,Ra^{3/2}E^{2}$ (3.10). For clarity, only the cases with $E\leqslant 10^{-3}$ and $\widetilde{Ra}<200$ are displayed in this figure. The symbols have the same meaning as in figure 1.

The plane layer numerical simulations of rotating convection by King et al. (Reference King, Stellmach and Aurnou2012) and Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) as well as the asymptotically reduced theoretical models by Julien et al. (Reference Julien, Knobloch, Rubio and Vasil2012a ) indicate that the convective heat flux only depends on the supercriticality $Ra/Ra_{c}$ when $Ro_{c}\ll 1$ . To check whether our numerical calculations actually support such a behaviour at low Ekman numbers, figure 5(a) shows $Nu$ as a function of $\widetilde{Ra}=RaE^{4/3}$ , since $Ra_{c}\sim E^{-4/3}$ . For $\widetilde{Ra}<6$ , $Nu$ slowly increases with the Rayleigh number and follows the previously described weakly nonlinear regime. Beyond this point, a fast steepening of the slope is observed and the function $Nu=f(\widetilde{Ra})$ starts to show an additional dependence on $E$ : while the large Ekman number cases quickly depart from a pure function of $\widetilde{Ra}$ , the low Ekman number simulations continue to follow a law of the form $Nu=f(\widetilde{Ra})$ up to higher $\widetilde{Ra}$ values. The exact location of this departure point gradually shifts to increasing $\widetilde{Ra}$ with decreasing $E$ . Figure 5(b) shows the compensated scalings of $Nu$ and reveals that the steepest part of the heat transfer law approaches the diffusivity-free scaling $Nu=0.15\widetilde{Ra}^{3/2}$ , at least in the range $6\leqslant \widetilde{Ra}\leqslant 20$ . To further assess the validity domain of the diffusivity-free scaling (1.3) in our numerical dataset, we introduce the local effective exponents $\unicode[STIX]{x1D6FC}_{eff}$ and $\unicode[STIX]{x1D6FD}_{eff}$ of the $Nu(Ra,E)$ law:

The local exponent $\unicode[STIX]{x1D6FD}_{eff}$ is unfortunately difficult to constrain since our dataset only samples sparse variations of the Ekman number (half-decade sampling). However, the dense coverage in $Ra$ for each Ekman number subset allows to estimate the variations of $\unicode[STIX]{x1D6FC}_{eff}$ . Figure 6 shows isocontours of $\unicode[STIX]{x1D6FC}_{eff}$ in a $(1/E,\widetilde{Ra})$ -plane. Beyond the weakly nonlinear regime delimited by the horizontal solid black line, the effective slope gradually increases with decreasing $E$ . It reaches a maximum value $\unicode[STIX]{x1D6FC}_{eff}\simeq 3/2$ for a small region around $E\lesssim 10^{-6}$ and $\widetilde{Ra}\simeq 10$ .

Figure 6.

Isocontours of the local effective exponent $\unicode[STIX]{x1D6FC}_{eff}$ of the $Nu=Ra^{\unicode[STIX]{x1D6FC}_{eff}}E^{\unicode[STIX]{x1D6FD}_{eff}}$ scaling in the $(1/E,\widetilde{Ra})$ -plane. The symbol shape corresponds to the Ekman number as in the previous figures, while the symbol colour scales here with the value of local slope $\unicode[STIX]{x1D6FC}_{eff}$ . The solid black line corresponds to the upper limit of the weakly nonlinear regime $Ra=6\,Ra^{4/3}$ , while the dashed black line corresponds to the dynamical boundary $Ra=0.4\,E^{-8/5}$ derived in figure 7.

The numerical simulations carried out by Cheng et al. (Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015) in Cartesian coordinates revealed a continuous increase of the local exponent $\unicode[STIX]{x1D6FC}_{eff}$ with decreasing $E$ and exhibit much steeper scaling beyond $\unicode[STIX]{x1D6FC}_{eff}\simeq 3$ (see also Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Kunnen et al. Reference Kunnen, Ostilla-Mónico, van der Poel, Verzicco and Lohse2016). This phenomenon has been attributed to the sizeable role of Ekman pumping that enhances the heat transfer even in the low Ekman number regime (Julien et al. Reference Julien, Aurnou, Calkins, Knobloch, Marti, Stellmach and Vasil2016). This steep scaling is not observed in absence of Ekman boundary layers, for instance when stress-free mechanical boundary conditions are adopted (Barker et al. Reference Barker, Dempsey and Lithwick2014; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014; Kunnen et al. Reference Kunnen, Ostilla-Mónico, van der Poel, Verzicco and Lohse2016; Plumley et al. Reference Plumley, Julien, Marti and Stellmach2016). In that case, local exponents close to $\unicode[STIX]{x1D6FC}_{eff}\simeq 3/2$ are recovered at low Ekman numbers.

The scaling exponents obtained in our simulations in spherical geometry for comparable Ekman numbers $E=O(10^{-7})$ remain bounded by the diffusivity-free scaling $\unicode[STIX]{x1D6FC}_{eff}=3/2$ (see also King et al. Reference King, Soderlund, Christensen, Wicht and Aurnou2010). Although a further increase of $\unicode[STIX]{x1D6FC}_{eff}$ for $E<3\times 10^{-7}$ cannot be a priori ruled out, the compensated scaling shown in figure 5(b) suggests a gradual approach to $\unicode[STIX]{x1D6FC}=3/2$ , which seems to leave little room for further increase. This suggests that our simulations approach the asymptotic diffusivity-free behaviour for the parameter range $E\leqslant 10^{-6}$ and $6\leqslant \widetilde{Ra}\leqslant 20$ .

Clearly however, the validity domain of the diffusivity-free scaling (1.3) remains relatively narrow in our set of numerical models. Beyond the steep scaling with $\unicode[STIX]{x1D6FC}_{eff}\simeq 3/2$ , our numerical calculations show a transition to a shallower heat transfer scaling at larger supercriticality $\widetilde{Ra}$ . The characterisation of the upper bound of the rotation-dominated regime of rotating convection has been a long-standing question. The ratio of the global-scale Coriolis and buoyancy forces, approximated by the value of $Ro_{c}$ , has long been postulated to control the transition from rotation-dominated convection to non-rotating convection (e.g. Gilman Reference Gilman1977; Zhong & Ahlers Reference Zhong and Ahlers2010; Stevens, Clercx & Lohse Reference Stevens, Clercx and Lohse2013). Using a combination of laboratory experiments and local numerical simulations in Cartesian coordinates, King et al. (Reference King, Stellmach, Noir, Hansen and Aurnou2009) however demonstrated that a simple $Ro_{c}$ criterion does not correctly capture the regime transition, which happens while the bulk Rossby number is still much smaller than unity (see also Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015). They instead suggested that this transition is controlled by the competing thickness of the thermal and Ekman boundary layers. The resulting cross-over was found to follow $RaE^{3/2}\sim 1$ (King et al. Reference King, Stellmach and Aurnou2012). However, numerical experiments that employed stress-free boundary conditions, in which viscous boundary layers are not present, yield a similar regime transition, which questions the role of the Ekman layer on the regime change (e.g. Schmitz & Tilgner Reference Schmitz and Tilgner2009).

Figure 7.

Nusselt number compensated by $Ra^{3/2}E^{2}$ as a function of $RaE^{8/5}$ . Only the cases with $E\leqslant 3\times 10^{-4}$ and $Nu\geqslant 2$ are displayed in this figure. The solid black line corresponds to the diffusivity-free scaling $Nu=0.15\,Ra^{3/2}E^{2}$ (3.10) already displayed in figure (5), while the vertical line corresponds to the dynamical boundary scaling $RaE^{8/5}=0.4$ . The symbols have the same meaning as in figure 1.

Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b ) thus argued that the transition from rotationally constrained to what they call ‘rotationally influenced’ convection is rather controlled by the dynamics of the thermal boundary layers. In particular, they proposed that the diffusion-free scaling (1.3) breaks down when the thermal boundary layer is no longer in geostrophic balance. In other words, $Nu\sim \widetilde{Ra}^{3/2}$ would hold as long as the local convective Rossby number in the thermal boundary layer $Ro_{\unicode[STIX]{x1D706}}$ remains smaller than unity. The definition of $Ro_{\unicode[STIX]{x1D706}}$ leads to

where $\unicode[STIX]{x1D706}$ is the thermal boundary layer thickness. Since

using (1.3), $Ro_{\unicode[STIX]{x1D706}}\sim 1$ thus yields

To test the applicability of this transition parameter, figure 7 shows the heat transfer data normalised by the diffusivity-free scaling (1.3), $NuRa^{-3/2}E^{-2}$ , versus $RaE^{8/5}$ for the numerical models with $E\leqslant 3\times 10^{-4}$ and $Nu\geqslant 2$ . The numerical simulations that fulfil $RaE^{8/5}<0.4$ lie very close to the geostrophic scaling (1.3). A best fit to the 16 cases that satisfy this criterion indeed yields $NuRa^{-3/2}E^{-2}=0.149(\pm 0.002)$ . The combination of the two criteria $RaE^{4/3}>6$ and $RaE^{8/5}<0.4$ defines a wedge in the parameter space highlighted by the two black lines visible in figure 6 where the scaling

holds. The transition parameter (3.9) introduced by Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b ) thus enables us to successfully demarcate the rotation-dominated regime and to efficiently collapse the heat transfer data across the regime transition.

3.2.2 Reynolds number scaling

To establish a scaling relation for the typical convective flow speed $Re_{c}$ , we examine first the variations of the viscous dissipation rate $\unicode[STIX]{x1D716}_{U}$ . In the limit of vanishing Ekman number, the viscous dissipation is expected to primarily occur in the bulk of the fluid (Busse Reference Busse1970). However, in the parameter range currently accessible to numerical models (i.e. $E\geqslant O(10^{-7})$ ), dissipation by friction through the Ekman boundary layers might still contribute substantially to the total viscous dissipation rate (e.g. Aubert et al. Reference Aubert, Brito, Nataf, Cardin and Masson2001). We thus follow the same strategy that has been fruitfully introduced by Grossmann & Lohse (Reference Grossmann and Lohse2000) for non-rotating RBC and decompose the viscous dissipation rate into a fluid bulk contribution $\unicode[STIX]{x1D716}_{U}^{bu}$ and a boundary layer contribution $\unicode[STIX]{x1D716}_{U}^{bl}$ :

where the integration domain is defined by ${\mathcal{T}}=\{r_{i}+\unicode[STIX]{x1D706}_{U}^{i}\leqslant r\leqslant r_{o}-\unicode[STIX]{x1D706}_{U}^{o};0\leqslant \unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x03C0};0\leqslant \unicode[STIX]{x1D719}\leqslant 2\unicode[STIX]{x03C0}\}$ , where $\unicode[STIX]{x1D706}_{U}^{i}$ ( $\unicode[STIX]{x1D706}_{U}^{o}$ ) are the thicknesses of the inner (outer) viscous boundary layer. We then establish scaling relations for the two contributions $\unicode[STIX]{x1D716}_{U}^{bu}$ and $\unicode[STIX]{x1D716}_{U}^{bl}$ as functions of $Re_{c}$ and $E$ , starting with the kinetic dissipation rate in the bulk of the fluid which can be estimated by

where $\ell _{diss}$ is the viscous dissipation length scale defined such that the local Reynolds number associated with this length scale is unity, i.e. $u_{diss}\,\ell _{diss}/\unicode[STIX]{x1D708}\sim 1$ . Following Davidson (Reference Davidson2013), we also introduce a small-scale vorticity $\unicode[STIX]{x1D714}_{diss}$ defined by $\unicode[STIX]{x1D714}_{diss}\sim u_{diss}/\ell _{diss}$ , which yields

At this stage, the study of the viscous dissipation rate needs to be complemented by a force balance analysis in order to estimate the scaling of $\unicode[STIX]{x1D714}_{diss}$ . A viscosity-free scaling of rotating convection was first proposed by Stevenson (Reference Stevenson1979) and Ingersoll & Pollard (Reference Ingersoll and Pollard1982) and later further developed by Aubert et al. (Reference Aubert, Brito, Nataf, Cardin and Masson2001). This scaling, known as the inertial scaling of rapidly rotating convection, hypothesises a triple force balance between Coriolis, inertia and Archimedean force (hereafter CIA). Taking the curl of the Navier–Stokes equation (2.2) in its dimensional form, the CIA triple balance yields

where the factor two in the vortex stretching term has been omitted. We now introduce the two integral length scales $\ell _{\bot }$ and $\ell _{/\!/}$ , respectively perpendicular and parallel to the rotation axis. Since rapidly rotating convective patterns take the form of columnar structures aligned with the rotation axis, $\ell _{\bot }\ll \ell _{/\!/}$ . Denoting $U_{c}$ the typical velocity, $\unicode[STIX]{x1D6E9}$ the typical thermal perturbation and $\unicode[STIX]{x1D714}_{c}$ the typical vorticity, one gets

We now assume that the axial dimension of the convection rolls $\ell _{/\!/}$ can be approximated by the container length scale $L$ . This assumption, combined with the balance between Coriolis force and inertia in (3.15), yields

We further assume that the convective heat flux per unit area ${\mathcal{Q}}$ scales as

which implies that the correlation between thermal perturbation and velocity are independent of $E$ , $Pr$ and $Ra$ (Gillet & Jones Reference Gillet and Jones2006). The balance between inertia and buoyancy in (3.15) then leads to

Since on time average $(\unicode[STIX]{x1D708}^{3}/L^{4})\,\unicode[STIX]{x1D716}_{U}\sim Ra_{{\mathcal{Q}}}Pr^{-2}$ , it follows from (3.13) $\unicode[STIX]{x1D714}_{diss}\sim \unicode[STIX]{x1D714}_{c}$ . The proportionality relation between the typical time scale at the integral length $\ell _{\bot }$ and at the dissipation scale $\ell _{diss}$ is a distinctive feature of two-dimensional turbulence (e.g. Davidson Reference Davidson2015). Finally assuming that $\unicode[STIX]{x1D714}_{c}\sim U_{c}/\ell _{\bot }$ , one can derive the following expression for the viscous dissipation rate in the fluid bulk

Figure 8.

(a) Dimensionless viscous dissipation rate $\tilde{\unicode[STIX]{x1D716}}_{U}$ as a function of the Reynolds number $Re_{c}$ . (b) Corresponding compensated $\tilde{\unicode[STIX]{x1D716}}_{U}$ scaling. The solid black lines correspond to the theoretical scaling derived in (3.22) with $a=0.066$ and $b=4.843$ . Only the cases that fulfil the conditions $Nu>2$ and $RaE^{8/5}<1$ have been selected on this figure. The error bars illustrate the r.m.s. fluctuations and correspond to one standard deviation from the time averages. The symbols have the same meaning as in figure 1.

The same procedure can be applied to the viscous dissipation in the Ekman boundary layers with the difference that the typical length scale is now the boundary layer thickness $\unicode[STIX]{x1D706}_{U}$ . Furthermore, the dissipation occurs only in the fraction of the fluid volume occupied by the viscous boundary layers. In the limit of $\unicode[STIX]{x1D706}_{U}\ll L$ , this volume fraction can be approximated by the ratio $\unicode[STIX]{x1D706}_{U}/L$ , thus neglecting any curvature effect. This yields

Since $\unicode[STIX]{x1D706}_{U}/L\sim E^{1/2}$ (see below in § 3.2.5), the viscous dissipation by friction through the boundary layers can be expressed by

Combining (3.19) and (3.21) allows us to derive the following scaling behaviour for the dimensionless viscous dissipation rate

To test the validity of this scaling, we calculate $\tilde{\unicode[STIX]{x1D716}}_{U}$ and $Re_{c}$ in our numerical simulations and directly fit the expression (3.22). This leaves only the two prefactors $a$ and $b$ as free fitting parameters. Figure 8(a) shows $\tilde{\unicode[STIX]{x1D716}}_{U}E^{1/2}$ as a function of $Re_{c}$ for the 41 cases that fulfil $Nu>2$ and $RaE^{8/5}<1$ . The best fit to the numerical data yields $a=0.066(\pm 0.003)$ and $b=4.843(\pm 0.096)$ . The expression (3.22) provides an excellent agreement with the data, further confirmed by the bulk compensated scaling $\tilde{\unicode[STIX]{x1D716}}_{U}E^{1/2}Re_{c}^{-5/2}$ shown in figure 8(b). Explicitly including the viscous friction that occurs in the Ekman boundary layers thus allows to very accurately describe the numerical data over a broad range of parameter $3\times 10^{-4}\leqslant E\leqslant 3\times 10^{-7}$ , $10^{2}\leqslant Re_{c}\leqslant 10^{4}$ . We also note that a stricter restriction to the 16 turbulent cases that satisfy the criterion $RaE^{8/5}<0.4$ and thus lie within the wedge displayed in figure (6) yields very similar values for $a$ and $b$ . The expression (3.22) also allows to estimate the transition value of $Re_{c}$ beyond which the viscous dissipation that occurs in the interior of the fluid will dominate the dissipation in boundary layers. Figure 8(b) demonstrates that our dataset nearly reaches this point where the bulk scaling alone provides a good description. Using the best fitting parameters $a$ and $b$ yields $Re_{c}=(b/a)^{2}\simeq 5500$ . Beyond this value, the viscous dissipation in the Ekman layers becomes secondary and the dynamical behaviour will then gradually tend towards the inertial scaling of rotating convection.

Figure 9.

(a) Reynolds number $Re_{c}$ as a function of $Ra_{{\mathcal{Q}}}Pr^{-2}E^{1/2}$ . (b) Corresponding compensated $Re_{c}$ scaling. The solid black lines correspond to the theoretical scaling derived in (3.26), while the dashed black lines correspond to the least-square fit to the data. Only the cases that fulfil the conditions $Nu>2$ and $RaE^{8/5}<1$ have been selected on this figure. The error bars illustrate the r.m.s. fluctuations and correspond to one standard deviation from the time averages. The symbols have the same meaning as in figure 1.

Assuming that the viscous dissipation either entirely occurs in the fluid bulk (i.e. $\unicode[STIX]{x1D716}_{U}\simeq \unicode[STIX]{x1D716}_{U}^{bu}$ ) or in the Ekman layers (i.e. $\unicode[STIX]{x1D716}_{U}\simeq \unicode[STIX]{x1D716}_{U}^{bl}$ ) allows one to define the two end-member scaling relations:

where

and

We note that (3.25) corresponds to the convective velocity scaling classically obtained in the CIA triple balance (e.g. Aubert et al. Reference Aubert, Brito, Nataf, Cardin and Masson2001; Gillet & Jones Reference Gillet and Jones2006; Barker et al. Reference Barker, Dempsey and Lithwick2014), while the scaling (3.24) has the same dependence on $Ra_{{\mathcal{Q}}}$ as the VAC triple balance (3.5) but carries a different $E$ -scaling exponent. Once $a$ and $b$ have been determined, the equation (3.22) can be numerically solved to derive the combined scaling law for $Re_{c}$ :

For comparison purposes, we also compute a simple best fit to the data which yields $Re_{c}=0.708(\pm 0.017)(Ra_{{\mathcal{Q}}}Pr^{-2}E^{1/2})^{0.473\pm 0.002}$ . Figure 9(a) shows $Re_{c}$ versus $Ra_{{\mathcal{Q}}}Pr^{-2}E^{1/2}$ for the 41 cases already shown in figure 8, while figure 9(b) shows the corresponding bulk compensated scaling.

Previous parameter studies of rotating convection in spherical shells either carried out under the quasi-geostrophic assumption (Gillet & Jones Reference Gillet and Jones2006; Guervilly Reference Guervilly2010) or in fully three-dimensional models (Christensen Reference Christensen2002; King & Buffett Reference King and Buffett2013) obtained steeper exponents than the $2/5$ scaling expected in the classical CIA balance (3.25). This discrepancy has usually been attributed to the significant role played by viscosity in numerical models and thus prompted several authors to rather describe their dataset using the VAC scaling hypothesis (3.5). Since the scaling exponents are relatively close to each other, $2/5$ versus $1/2$ regarding the dependence on $Ra_{{\mathcal{Q}}}$ , discriminating one scaling from another remains however a difficult task (King & Buffett Reference King and Buffett2013). (Note however that the Ekman number dependence is not the same for these two scalings.) Here we show that the separation of $\unicode[STIX]{x1D716}_{U}$ into boundary layers and bulk contributions allows us to very accurately describe the numerical data. Moreover, in the range of $Re_{c}$ covered by our numerical dataset (i.e. $10^{2}\leqslant Re_{c}<10^{4}$ ), the simple power law $Re_{c}=0.708(\pm 0.017)(Ra_{{\mathcal{Q}}}Pr^{-2}E^{1/2})^{0.473\pm 0.002}$ provides a statistically nearly indiscernible fit quality from the scaling theory (3.26). Deviations only become more significant at the high end of the parameter range, where the power law fails to accurately capture the complex dependence of $Re_{c}$ upon $Ra_{{\mathcal{Q}}}Pr^{-2}E^{1/2}$ . This explains why previous analyses that reduced the scaling behaviours of $Re_{c}$ to such simple power laws obtained scaling exponents robustly larger than $2/5$ (e.g. Gillet & Jones Reference Gillet and Jones2006; King & Buffett Reference King and Buffett2013).

Figure 10.

(a) Average flow length scale $\ell /L$ calculated using (2.14) as a function of the Ekman number for the numerical models with $E\leqslant 3\times 10^{-4}$ and $Nu>2$ . For comparison, the weakly nonlinear scaling obtained in figure 4 has been plotted as a solid black line. (b) Corresponding compensated $\ell /L$ scaling. Only the 41 cases that satisfy $Nu>2$ and $RaE^{8/5}<1$ have been selected on this figure. The solid black line corresponds to the best fit to the cases with $E=3\times 10^{-7}$ . The error bars illustrate the r.m.s. fluctuations and correspond to one standard deviation from the time averages. The symbols have the same meaning as in figure 1.

3.2.3 Flow length scales and interior temperature gradients

Using the equations (3.16) and (3.25), the inertial theory of rotating convection predicts the following scaling for the integral flow length scale $\ell _{\bot }$

which is generally referred to as the Rhines scaling (Rhines Reference Rhines1975). Using the numerical models computed by Christensen & Aubert (Reference Christensen and Aubert2006), several analyses of the typical flow length scale suggest that the VAC scaling (3.4) might possibly hold beyond the weakly nonlinear regime of rotating convection (e.g. King & Buffett Reference King and Buffett2013; Oruba & Dormy Reference Oruba and Dormy2014). Figure 10(a) shows the calculations of the average flow length scale $\ell /L$ plotted as a function of $E$ for the numerical models with $E\leqslant 3\times 10^{-4}$ and $Nu>2$ . The $\ell \sim E^{1/3}\,L$ scaling that accurately describes the numerical data close to onset of convection (see figure 4) fails to properly capture the length scale variations in the nonlinear regime. Especially, at low Ekman numbers ( $E\leqslant 3\times 10^{-6}$ ), $\ell$ exhibits a strong additional dependence on the convective forcing and increases with the supercriticality (see table 2 for details). Figure 10(b) shows the calculations of $\ell$ versus $Re_{c}E$ for the 41 numerical models already shown in figures 8 and 9. A best fit to the cases with $E=3\times 10^{-7}$ that fulfil $RaE^{8/5}<0.4$ yields $\ell /L=1.016(\pm 0.212)(Re_{c}E)^{0.45(\pm 0.027)}$ , reasonably close to the theoretical scaling (3.27). Since the dissipation that occurs in the boundary layers still plays a substantial role in the velocity scaling (3.22), it is not surprising that (3.27) is only approached for the lowest Ekman number considered in this study.

Figure 11.

(a) Radial profile of the time and horizontally averaged temperature $\unicode[STIX]{x1D717}(r)$ for different convective Rossby number $Ro_{c}$ with a fixed Ekman number of $E=10^{-4}$ . The dot-dashed black line corresponds to the conductive temperature profile solution of (2.10). (b) Radial profile of the time and horizontally averaged temperature $\unicode[STIX]{x1D717}(r)$ for different Ekman numbers with approximately the same Nusselt number $10.3<Nu<11$ .

While non-rotating convection undergoes an efficient turbulent mixing which results in an isothermal fluid interior (e.g. Verzicco & Camussi Reference Verzicco and Camussi1999), the prominent role played by the Coriolis force in rotating convection impedes the mixing. Both experiments (e.g. Boubnov & Golitsyn Reference Boubnov and Golitsyn1990; Kunnen et al. Reference Kunnen, Geurts and Clercx2010) and direct numerical simulations (e.g. Julien et al. Reference Julien, Legg, McWilliams and Werne1996; King et al. Reference King, Stellmach and Aurnou2012; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) have indeed revealed that a large-scale interior temperature gradient can be maintained in the rapidly rotating regime (see also Gillet & Jones Reference Gillet and Jones2006; King et al. Reference King, Soderlund, Christensen, Wicht and Aurnou2010). Figure 11 shows the time- and horizontally averaged temperature profile $\unicode[STIX]{x1D717}(r)$ for several rotating and non-rotating models. At a fixed Ekman number, an increase of the supercriticality (figure 11 a) changes the temperature distribution from a nearly conductive state (solid orange line) to a nearly isothermal fluid bulk (dot-dashed line). At larger $Ro_{c}$ (broken lines), the rotational constraint decreases and thin thermal boundary layers start to emerge and tend to accommodate an increasing fraction of the temperature drop. As visible in figure 11(b), the transition from rotating to non-rotating convection is thus associated with a gradual lowering of the average temperature gradient and a transition to a physical regime where boundary layers entirely control the heat transfer.

Figure 12.

(a) Temperature gradient at mid-shell radius $\unicode[STIX]{x1D6FD}_{T}$ versus $\widetilde{Ra}$ for the numerical models with $Nu>2$ and $E\leqslant 10^{-3}$ . (b) Absolute temperature gradient at mid-shell renormalised by $E^{-1/2}$ versus $NuRe_{c}^{-3/2}$ for the numerical models with $RaE^{8/5}<1$ and $Nu>2$ . The solid black line corresponds to the best fit to the data. The symbols have the same meaning as in figure 1.

Measuring the interior temperature gradient $\unicode[STIX]{x1D6FD}_{T}$ (2.13a,b ) thus helps to characterise the heat transfer regime. Figure 12(a) shows $\unicode[STIX]{x1D6FD}_{T}$ versus $\widetilde{Ra}$ for the set of numerical simulations with $E\leqslant 10^{-3}$ . An increase of the supercriticality $\widetilde{Ra}$ leads to a gradual decrease of $|\unicode[STIX]{x1D6FD}_{T}|$ . The thermal profiles in the fluid bulk tend to a thermally well-mixed interior (i.e. vanishing $\unicode[STIX]{x1D6FD}_{T}$ ) when the rotational influence on the flow decreases. In contrast to the asymptotic calculations by Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b ), our dataset does not show any evidence for a direct scaling relation between $\unicode[STIX]{x1D6FD}_{T}$ and $\widetilde{Ra}$ . However, following Stevenson (Reference Stevenson1979) and Barker et al. (Reference Barker, Dempsey and Lithwick2014), the interior temperature gradient can be estimated by

in the rapidly rotating limit. Using (3.27) for the length scale scaling and approximating $\unicode[STIX]{x1D6E9}/\unicode[STIX]{x0394}T$ by $Nu/Re_{c}$ then yields

Figure 12(b) shows that this scaling provides a reasonable description of the 41 numerical cases already discussed in figures 8 and 9. The best fit to the data however gives a shallower exponent than expected from theory: $\unicode[STIX]{x1D6FD}_{T}E^{1/2}=0.643(\pm 0.027)(NuRe_{c}^{-3/2})^{0.812(\pm 0.005)}$ . This discrepancy is not surprising since the underlying $\ell _{\bot }$ scaling that enters (3.29) is only partially realised in our set of simulations (see figure 10). Lowering the Ekman number further might help to ascertain this scaling. Indeed, the local numerical calculations carried out in Cartesian coordinates by Barker et al. (Reference Barker, Dempsey and Lithwick2014) demonstrate the validity of (3.29) once the influence of the viscous boundary layers has been minimised. Replacing the expected asymptotic scalings for $Nu$ (1.3) and $Re_{c}$ (3.25) in (3.29) yields $\unicode[STIX]{x1D6FD}_{T}\sim Pr$ . This implies that $\unicode[STIX]{x1D6FD}_{T}$ should saturate on a constant value in the turbulent quasi-geostrophic regime of rotating convection. Local numerical calculations carried out in Cartesian coordinates by Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) indeed obtain a saturation of $\unicode[STIX]{x1D6FD}_{T}$ for $E=O(10^{-7})$ and $\widetilde{Ra}\sim 100$ .

Figure 13.

(a,b) Thermal boundary layer thickness at the inner (outer) boundary $\unicode[STIX]{x1D706}_{T}^{i}$ ( $\unicode[STIX]{x1D706}_{T}^{o}$ ) as a function of the Nusselt number. The solid black lines correspond to the theoretical scalings for non-rotating convection derived in Gastine et al. (Reference Gastine, Wicht and Aurnou2015). (c,d) Compensated $\unicode[STIX]{x1D706}_{T}^{i}/L$ ( $\unicode[STIX]{x1D706}_{T}^{o}/L$ ) scaling. The symbols have the same meaning as in figure 1.

3.2.4 Thermal boundary layers

Since the heat transport in the thermal boundary layers is controlled by diffusion, the Nusselt number provides an estimate for the ratio between the thermal boundary layer thickness and the related temperature drop:

where $\unicode[STIX]{x1D706}_{T}^{i}$ ( $\unicode[STIX]{x1D706}_{T}^{o}$ ) is the thickness of the thermal boundary layer at the inner (outer) boundary and $\unicode[STIX]{x0394}T_{i}$ ( $\unicode[STIX]{x0394}T_{o}$ ) is the associated temperature contrast. This yields the following scalings for the thermal boundary layer thicknesses at both boundaries:

In non-rotating convection in spherical geometry, the thermal boundary layers are asymmetric and thus $\unicode[STIX]{x0394}T^{i}/\unicode[STIX]{x0394}T^{o}\neq 1$ . In a previous study, we demonstrated that the ratio $\unicode[STIX]{x0394}T^{i}/\unicode[STIX]{x0394}T^{o}$ only depends on the radius ratio $\unicode[STIX]{x1D702}$ and on the gravity profile $\tilde{g}(r)$ (Gastine et al. Reference Gastine, Wicht and Aurnou2015). Thus, the scaling for the boundary layer thickness in non-rotating RBC is

where only the scaling prefactors depend on the geometry and on the gravity distribution of the spherical shell. In rotating convection however, the temperature drops at each thermal boundary layer are likely to also depend on $Ra$ and $E$ and hence a non-trivial additional dependence on $Nu$ might be expected (see figure 11). Figure 13 shows the calculations of $\unicode[STIX]{x1D706}_{T}^{i}$ and $\unicode[STIX]{x1D706}_{T}^{o}$ using the slope-intersection method described in § 2.3 versus $Nu$ for both rotating and non-rotating simulations. As pointed out by Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b ), other methods to estimate the thermal boundary layers in rotating convection might yield different thicknesses. A comparison between the values obtained from the slope-intersection method with the location of the peaks of the r.m.s. of the temperature fluctuations for a few selected cases however yield similar boundary layer thicknesses (for a comparison, see Gastine et al. Reference Gastine, Wicht and Aurnou2015). Large Ekman number models ( $E\geqslant 10^{-2}$ ) essentially behave similarly to the non-rotating cases. At the inner boundary of the spherical shell, the cases with intermediate Ekman numbers $3\times 10^{-5}\leqslant E\leqslant 3\times 10^{-3}$ exhibit a lower prefactor and a slightly shallower slope than $Nu^{-1}$ particularly visible in the compensated plot displayed in figure 13(c). The cases with $E\leqslant 3\times 10^{-6}$ , however seem to follow again a scaling $\unicode[STIX]{x1D706}_{T}^{i}/L\sim Nu^{-1}$ , though with a much smaller prefactor ( $\simeq 0.17$ instead of $\simeq 0.4$ ) than the non-rotating cases. The thermal boundary layer at the outer boundary behaves quite differently. On average, $\unicode[STIX]{x1D706}_{T}^{o}$ remains much closer to the values found in the non-rotating models and can thus be roughly approximated by $\unicode[STIX]{x1D706}_{T}^{o}/L\sim Nu^{-1}$ . However, a secondary, intricate dependence on the Ekman number remains, as visible on the compensated scaling of $\unicode[STIX]{x1D706}_{T}^{o}$ displayed in figure 13(d). For both boundary layers, the $Nu^{-1}$ scaling thus provides an acceptable first-order approximation of the thermal boundary layer thickness measured by the slope-intersection method, though subtle dependences on the Ekman number can impinge on both the prefactor and the scaling exponent.

3.2.5 Viscous boundary layers

In the range of Rayleigh numbers explored here ( $10^{3}\leqslant Ra\leqslant 10^{11}$ ), the viscous boundary layers in non-rotating convection are usually assumed to be laminar and follow the Prandtl–Blasius boundary layer theory (Prandtl Reference Prandtl1905; Blasius Reference Blasius1908). Balancing the inertia of the fluid bulk with the viscous forces in the boundary layers yields

In contrast, owing to the primary role of Coriolis force, the viscous boundary layers in rotating convection, called the Ekman layers, differ greatly from non-rotating convection. A scaling relation for the Ekman layers can be derived by considering a force balance between Coriolis force and viscosity in the limit of $E\rightarrow 0$ (e.g. Greenspan Reference Greenspan1968):

Figure 14 shows the calculations of the viscous boundary layer thicknesses for both the inner and the outer spherical shell boundaries as a function of $Re_{c}$ and $E$ . Here we only examine the mean viscous boundary layer, though in spherical geometry the Ekman layer thickness might also depend on the latitude (Greenspan Reference Greenspan1968). When the Ekman number is large ( $E\geqslant 3\times 10^{-3}$ ), the influence of rotation on the viscous boundary layer remains secondary. $\unicode[STIX]{x1D706}_{U}^{i}$ and $\unicode[STIX]{x1D706}_{U}^{o}$ are then essentially the same as the non-rotating cases (figure 14 a,b). As already shown in our previous study (Gastine et al. Reference Gastine, Wicht and Aurnou2015), when $Re_{c}\geqslant 250$ , the scaling exponents are in close agreement with the Prandtl–Blasius scaling (3.33). At lower Ekman numbers ( $E\leqslant 3\times 10^{-5}$ ), the viscous boundary layer thicknesses become almost independent of $Re_{c}$ and approach the theoretical scaling for Ekman layers (figure 14 c,d). Indeed, the best fit to the numerical cases that fulfil $RaE^{8/5}<1$ and $Nu>2$ yields $\unicode[STIX]{x1D706}_{U}^{i}/L=0.875(\pm 0.084)\,E^{0.465(\pm 0.008)}$ and $\unicode[STIX]{x1D706}_{U}^{o}/L=1.109(\pm 0.042)\,E^{0.497(\pm 0.003)}$ , in good agreement with (3.34).

Figure 14.

(a,b) Viscous boundary layer thickness at the inner (outer) boundary $\unicode[STIX]{x1D706}_{U}^{i}$ ( $\unicode[STIX]{x1D706}_{U}^{o}$ ) as a function of $Re_{c}$ . The solid lines correspond to the scalings obtained for non-rotating convection derived in Gastine et al. (Reference Gastine, Wicht and Aurnou2015). (c,d) Viscous boundary layer thickness at the inner (outer) boundary as a function of the Ekman number. The solid black lines correspond to the best fit to the 41 cases that fulfil $RaE^{8/5}<1$ and $Nu>2$ . The symbols have the same meaning as in figure 1.

3.3.1 Nusselt and Reynolds numbers

Figure 16 shows the calculations of the Nusselt number compensated by the heat transfer scaling for non-rotating convection as a function of the transition parameter (3.36). In figure 16(a), $Nu$ is compensated on the $y$ -axis by the asymptotic scaling $Nu_{NR}\sim Ra^{1/3}$ . Well into the rapidly rotating regime (i.e. $RaE^{12/7}\ll 1$ ), the heat transfer follows the $Nu\sim Ra^{3/2}E^{2}$ scaling (solid line). The growth of $RaE^{12/7}$ is then accompanied by a lowering of the heat transfer slope which levels off around the non-rotating asymptotic scaling when $RaE^{12/7}\gg 1$ (dotted vertical line). Similarly to King et al. (Reference King, Stellmach and Aurnou2012), the data are more scattered in the weakly rotating regime. This is expected since the asymptotic scaling $Nu_{NR}\sim Ra^{1/3}$ used to renormalise the data is not yet realised in the range of Rayleigh numbers covered by our numerical dataset. A much better collapse of the data can nevertheless be achieved by normalising $Nu$ with the actual calculations of $Nu_{NR}$ , the Nusselt numbers of the corresponding non-rotating simulations at the same Rayleigh numbers (figure 16 b). Since a direct computation of the non-rotating models with $Ra>10^{9}$ becomes numerically intractable, the scaling laws derived in Gastine et al. (Reference Gastine, Wicht and Aurnou2015) have been used to estimate $Nu_{NR}$ for the cases with $Ra>10^{9}$ . This renormalisation allows us to better describe the range $10\leqslant RaE^{12/7}\leqslant 10^{4}$ and to remarkably collapse all the rotating simulations with $Nu>2$ on one single curve.

Figure 17.

(a) Dimensionless viscous dissipation rate $\tilde{\unicode[STIX]{x1D716}}_{U}$ normalised by the scaling for the viscous dissipation rate obtained in non-rotating convection $\tilde{\unicode[STIX]{x1D716}}_{U}^{NR}=7.08\,Re_{c}^{5/2}+0.25\,Re_{c}^{3}$ (see Gastine et al. Reference Gastine, Wicht and Aurnou2015) as a function of $RaE^{12/7}$ (3.36). (b) Reynolds number $Re_{c}$ renormalised by $Ra^{1/2}$ as a function of $RaE^{12/7}$ . On both panels, the weakly nonlinear cases with $Nu<2$ have been excluded. The dotted vertical lines correspond to the Rayleigh number $Ra=100\,E^{-12/7}$ beyond which $\tilde{\unicode[STIX]{x1D716}}_{U}$ and $Re_{c}$ adopt scaling behaviours close to the non-rotating RBC. The symbols have the same meaning as in figure 1.

We can check whether the regime transition criterion (3.36) also applies to the scalings of the dimensionless viscous dissipation rate $\tilde{\unicode[STIX]{x1D716}}_{U}$ and of the flow speed $Re_{c}$ . In non-rotating convection, $\tilde{\unicode[STIX]{x1D716}}_{U}$ can also be decomposed into fluid bulk and boundary layer contributions. Grossmann & Lohse (Reference Grossmann and Lohse2000) predicted a scaling of the form $\tilde{\unicode[STIX]{x1D716}}_{U}^{NR}\sim a\,Re_{c}^{3}+b\,Re^{5/2}$ , which has been found to also accurately describe the numerical models of non-rotating convection in spherical shells (Gastine et al. Reference Gastine, Wicht and Aurnou2015). Figure 17(a) thus shows $\tilde{\unicode[STIX]{x1D716}}_{U}$ renormalised by $\tilde{\unicode[STIX]{x1D716}}_{U}^{NR}$ found by Gastine et al. (Reference Gastine, Wicht and Aurnou2015) as a function of the transition parameter. In non-rotating convection, the dependence of the Reynolds number upon $Ra$ cannot be simply reduced to a simple power law but rather exhibits continuous changes in the local slope exponent, analogous to the Nusselt number scaling. We thus follow the same procedure as for the Nusselt number and renormalise $Re_{c}$ by the expected asymptotic scaling from the Grossmann & Lohse (Reference Grossmann and Lohse2000) theory in the limit of large $Ra$ , i.e. $Re_{c}^{NR}\sim Ra^{1/2}$ . Figure 17(b) shows the calculations of $Re_{c}Ra^{-1/2}$ versus $RaE^{12/7}$ . Except for the intermediate parameter range $1\leqslant RaE^{12/7}\leqslant 10^{2}$ , the numerical data for both $\tilde{\unicode[STIX]{x1D716}}_{U}$ and $Re_{c}$ are well collapsed on a single curve. We note that the scatter of the data in (b) could be possibly further reduced by rather considering the actual values of $Re_{c}^{NR}$ calculated in the non-rotating models. Beyond $RaE^{12/7}=O(10^{2})$ (dotted vertical lines), both $\tilde{\unicode[STIX]{x1D716}}_{U}$ and $Re_{c}$ adopt scaling behaviours close to the classical non-rotating RBC.

3.3.2 Flow length scale and interior temperature gradient

We now turn to discussing the variations of the bulk temperature gradient $\unicode[STIX]{x1D6FD}_{T}$ and average flow length scale $\ell$ across the regime transition from rotating to non-rotating convection. Figure 18(a) shows $\unicode[STIX]{x1D6FD}_{T}$ as a function of $RaE^{12/7}$ . In contrast to the previous quantities, we observe a significant scatter of the data in the rapidly rotating regime. As discussed above, and similarly to the findings by King et al. (Reference King, Stellmach and Buffett2013), $\unicode[STIX]{x1D6FD}_{T}$ depends on $Ra$ and $E$ in a complex fashion that is not captured by the transition parameter (3.36). In the weakly rotating limit (i.e. $RaE^{12/7}>10^{2}$ ), the bulk of the fluid becomes thermally well mixed, leading to $\unicode[STIX]{x1D6FD}_{T}\simeq 0$ .

Figure 18.

(a) Temperature gradient at mid-shell $\unicode[STIX]{x1D6FD}_{T}$ as a function of $RaE^{12/7}$ (3.36). (b) Convective flow length scale $\ell$ normalised by the scaling for the plume spacing obtained in non-rotating convection $\ell _{NR}$ as a function of $RaE^{12/7}$ (Gastine et al. Reference Gastine, Wicht and Aurnou2015). On both panels, the weakly nonlinear cases with $Nu<2$ have been excluded. The dotted vertical lines correspond to the Rayleigh number $Ra=100\,E^{-12/7}$ beyond which $\unicode[STIX]{x1D6FD}_{T}$ and $\ell$ adopt scaling behaviours close to the non-rotating RBC. The symbols have the same meaning as in figure 1.

The study of the flow length scale necessitates the determination of an asymptotic scaling for $\ell$ in the non-rotating regime of convection. In classical RBC experiments with rigid sidewalls, the bulk flow is dominated by a large-scale circulation (LSC) pattern. The average flow length scale can therefore be approximated by the vertical size of the container $L$ . In spherical shells however, the absence of sidewalls precludes the formation of such a global and single cell LSC structure. The typical flow length scale can instead be estimated by the average inter-plume distance (King et al. Reference King, Stellmach and Buffett2013). In our previous study (Gastine et al. Reference Gastine, Wicht and Aurnou2015), we established the following scaling for the plume spacing

Figure 18(b) shows the calculations of $\ell$ renormalised by $\ell _{NR}$ versus the transition parameter $RaE^{12/7}$ . Though the collapse of the data, especially when $RaE^{12/7}<1$ , is not as good as those obtained for $Nu$ , $\tilde{\unicode[STIX]{x1D716}}_{U}$ and $Re_{c}$ , the transition parameter (3.36) provides an effective way to capture the transition of the mean flow length scale from rotating (3.27) to non-rotating (3.37) convection. For $RaE^{12/7}>100$ (dotted vertical line), $\ell$ approaches the non-rotating scaling $\ell _{NR}$ (3.37).

3.3.3 Thermal and viscous boundary layers

In figure 19 we examine calculations of the thermal and viscous boundary layer thicknesses at the inner boundary renormalised by their asymptotic scalings in non-rotating convection plotted versus $RaE^{12/7}$ . Once again, the renormalisation of the axes allows us to collapse the boundary layer thicknesses for all $E$ on one single curve. The transition parameter $RaE^{12/7}$ therefore provides an accurate way of distinguishing the variations of most of the physical quantities considered in this study across the regime change from rotating to non-rotating convection.

Beyond $RaE^{12/7}=O(10^{2})$ , all the diagnostic quantities for the rotating models becomes statistically indiscernible from the non-rotating behaviour. This value can therefore be adopted to define the upper bound of the transitional regime.

Figure 19.

(a) Thermal boundary layer thickness at the inner boundary ( $\unicode[STIX]{x1D706}_{T}^{i}$ ) normalised by $Nu^{-1}$ as a function of $RaE^{12/7}$ (3.36). The solid black line corresponds to the theoretical scaling obtained in Gastine et al. (Reference Gastine, Wicht and Aurnou2015) for non-rotating convection; (b) viscous boundary layer thickness at the inner boundary ( $\unicode[STIX]{x1D706}_{U}^{i}$ ) normalised by $Re_{c}^{-1/2}$ as a function of $RaE^{12/7}$ . The dotted vertical lines correspond to the Rayleigh number $Ra=100\,E^{-12/7}$ beyond which $\unicode[STIX]{x1D706}_{T}^{i}$ and $\unicode[STIX]{x1D706}_{U}^{i}$ adopt scaling behaviours close to the non-rotating RBC. The symbols have the same meaning as in figure 1.