Diffusion‐Free Scaling in Rotating Spherical Rayleigh‐Bénard Convection

Abstract Direct numerical simulations are employed to reveal three distinctly different flow regions in rotating spherical Rayleigh‐Bénard convection. In the high‐latitude region I vertical (parallel to the axis of rotation) convective columns are generated between the hot inner and the cold outer sphere. The mid‐latitude region II is dominated by vertically aligned convective columns formed between the Northern and Southern hemispheres of the outer sphere. The diffusion‐free scaling, which indicates bulk‐dominated convection, originates from this mid‐latitude region. In the equator region III, the vortices are affected by the outer spherical boundary and are much shorter than in region II.

planar convection, see Figure 1a, the rotation axis is orthogonal to the plates, and the convective columns are homogeneously distributed in the horizontal direction and always stretch between the hot and cold plates. However, in spherical geometry, the rotation effect is latitude dependent; see Figure 1b, due to which three distinctly different flow regions are formed. Inside the inner sphere's tangent cylinder, the convective columns touch the inner and outer spherical boundaries. In the mid-latitude region, the convective columns are stretched between the Northern and Southern hemispheres of the outer sphere. Near the equator, the convective columns adjust themselves to the curved boundary. This work will show that the diffusion-free scaling originates from this mid-latitude region. The article is organized as follows: In Section 2, we introduce the rotating spherical RB system with its control parameters. Section 3 is an overview of our simulation results compared and validated to literature, subsequent analysis is performed in Sections 4 and 5. Finally, we conclude our findings in Section 6.
i o E T T . No-slip boundary conditions are imposed at both spheres. We solve the Navier-Stokes equations in spherical coordinates within the Boussinesq approximation, which in dimensionless form read: T x t , and ( ) E g r denote the fluid velocity, pressure, temperature and radially dependent gravitational acceleration.
In this study, we focus on a radius ratio    r r i o / 0 6 . and the gravity profile g r r r o ( ) ( )  / 2 valid for homogeneous mass distribution to allow comparisons with non-rotating (Gastine et al., 2015) and rotating  convection in spherical RB. This system configuration is considered representative for studying convection in gas giants (Long et al., 2020). Additionally, we perform simulations for   0.35 E and g r r r o ( ) ( )   / 1 , which is considered an Earth-like configuration used by Long et al. (2020) and Yadav et al. (2016). The equations are discretized by a staggered central second-order finite-difference scheme in spherical coordinates (Santelli et al., 2020). We use a uniform grid in the longitudinal and co-latitudinal directions and ensure that the bulk and boundary layers are appropriately resolved (Stevens, Verzicco, & Lohse, 2010). The grid cells are clustered toward the inner and outer sphere to ensure the boundary layers are adequately resolved (Shishkina et al., 2010). Further details on the simulations are given in the Supporting Information S1.
The dynamics of rotating spherical RB convection are determined by the Rayleigh, Prandtl, and Ekman numbers: where  E is the thermal expansion coefficient, o E g is the gravity at the outer sphere,  E is the kinematic viscosity, and  E is the thermal diffusivity of the fluid. E Ra is a measure of the thermal driving of the system, E Ek characterizes the ratio of viscous to Coriolis forces, and E Pr indicates the ratio of the viscous to thermal diffusivities. In this study we consider  1 E Pr . We use the Rossby number Ro Ra Pr Ek  / /2 to evaluate the relative importance of rotation and buoyancy (Gilman, 1977). We normalize the results using the length scale   o i E d r r , the temperature difference Δ E T between inner and outer sphere, and the free-fall velocity The Nusselt number quantifies the non-dimensional heat transport:    . sphere (region I ) and close to the equator region along the outer sphere (region III E ). We determine the distribution of the thermal dissipation rate over the different regions as follows: T  T  T  T  T  T  I bulk  I TBL  II bulk  II TBL  III  is not determined based on the thermal dissipation profiles as there is not a clear peak in the direction separating the regimes. Therefore, as discussed above, we use the maximum in the zonal flow profile to determine this transition.  In the following section, we will show that, in agreement with theoretical expectations discussed above, the scaling of the heat transfer in the region II E follows the diffusion-free scaling for rotation dominated strongly thermally driven flows. Figure 5 shows E Nu on the outer sphere compensated with the diffusion-free scaling law. Panel 5(a) shows that for the global heat transfer and    5 5 10 E Ek the diffusion-free scaling is observed for  6 E R . The crossover from the quasi-geostrophic region to the transitional region is observed at

Diffusion-Free Scaling in Region II
Ek . Figures 5b-5d show the heat transfer scaling in the different flow regions identified above. Panel 5(b) evidences that, due to Ekman pumping (Stellmach et al., 2014;Stevens, Clercx, & Lohse, 2010;Stevens et al., 2013;Zhong et al., 2009), the heat transport scaling in region I is  2.1 I E Nu R . This is steeper than the   3 2 / scaling for diffusion-free convection, but shallower than the   3 E value observed in planar convection (King et al., 2013). Most importantly, panel 5(c) shows that the diffusion-free scaling is much more pronounced in region II E than in region I . Although the diffusion-free scaling still starts at  6 E R , it continues for much higher E R than the global heat transfer, see Figure 5a. Panel 5(d) shows that no diffusion-free scaling regime is observed in region III E .
The diffusion-free scaling regime is observed from 6 E R up to t E Ra , where t E Ra indicates the E Ra number at which the regime for bulk-limited heat transfer in geostrophic turbulence ends . It was demonstrated  that for the global heat transfer the diffusion-free scaling regime is observed up to Ek , see also Figure 6a. For region II E , Figure 6b shows that the diffusion-free scaling is observed up to Ek , which is considerably higher E Ra than for the global heat transport.
In Section 4 of the Supporting Information S1, we show that the observation of the diffusion-free scaling in the mid-latitude region II E does not depend on the specific   0.

Conclusions
In conclusion, we have shown that rotating spherical RB convection has three distinctly different flow regions; see Figure 3b. In region I , convective columns are formed between the hot inner and cold outer spheres. The mid-latitude region II E is the region where the vertically aligned vortices are strongest, and the flow is bulk dominated. Region III E is formed around the equator, and here the vortices are shorter and are affected by the outer spherical boundary.
The diffusion-free scaling Nu RaEk with   3 2 / originates from the mid-latitude flow region in which the flow dynamics are bulk dominated. In this region, thin and long convective columns are formed between the Northern and Southern parts of the cold outer sphere. This geostrophically dominated flow region can be formed due to the system geometry. Due to the curvature effects in spherical geometries, the latitude-dependent Coriolis force results in inhomogeneous convective columns in the co-latitudinal direction and more convective columns on the outer sphere than the inner sphere.

Data Availability Statement
The data used in this article are available for download at https://doi.org/10.5281/zenodo.5034407. W. and R. J. A. M. S. acknowledge the financial support from ERC (the European Research Council) Starting Grant No. 804283 UltimateRB. This work was sponsored by NWO Science for the use of supercomputer facilities. The authors also acknowledge the national e-infrastructure of SURFsara, a subsidiary of SURF cooperation, the collaborative ICT organization for Dutch education and research, and Irene at Très Grand Centre de Calcul du CEA (TGCC) under PRACE project 2019215098. We acknowledge PRACE for awarding us access to MareNostrum at Barcelona Supercomputing Center (BSC), Spain (Project 2020235589, 2020225335).