Turbulent Rotating Rayleigh–Bénard Convection

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INTRODUCTION TO ROTATING RAYLEIGH-BÉNARD CONVECTION
Rotating Rayleigh-Bénard convection (RRBC) (Chandrasekhar 1953, Nakagawa & Frenzen 1955, Veronis 1959, Rossby 1969, Lucas et al. 1983, Boubnov & Golitsyn 1986, Zhong et al. 1993, Boubnov & Golitsyn 1995, Julien et al. 1996, Knobloch 1998, Hart et al. 2002, Vorobieff & Ecke 2002, Stevens et al. 2013, Kunnen 2021) is considerably more complex than its nonrotating counterpart (Ahlers et al. 2009b).Instead of three dimensionless control parameters in classical Rayleigh-Bénard convection (RBC)-the Rayleigh number Ra, the Prandtl number Pr, and the container geometry, represented by its aspect ratio -there are two additional parameters in RRBC, the rotation rate in the Ekman number Ek (alternatively, the Taylor number T a or convective Rossby number Ro), and centrifugal acceleration in the Froude number Fr (see the sidebar titled Dimensionless Quantities in RRBC).Some of the complexity of RRBC comes from the presence of two linear instabilities associated with the onset of convection: one from the conductive state to convection in the bulk for a laterally infinite system (Chandrasekhar 1953) and another occurring at smaller Ra to a state of traveling wall modes localized near the vertical boundaries of the container (Rossby 1969, Buell & Catton 1983, Pfotenhauer et al. 1987, Zhong et al. 1991, Ecke et al. 1992).Another important feature of bulk RRBC is that the nonlinear state is subject to a nonlinear instability [e.g., the Küppers-Lortz instability (Küppers & Lortz 1969, Cox & Matthews 2000)] so that the observed states of bulk convection even near onset are always time dependent (e.g., Zhong et al. 1993, Ning & Ecke 1993).Finally, there is a regime of quasi-geostrophic convection and geostrophic turbulence [Boubnov & Golitsyn 1986;Sakai 1997;Sprague et al. 2006;Julien et al. 2012a,b;Kunnen 2021] where in the limit of rapid rotation the Coriolis force is dominantly balanced by the pressure gradient.This regime is associated with astrophysical and geophysical flows (Pedlosky 1987, Glatzmaier 2014).
RRBC has enjoyed a renaissance in the past decade, with numerous advances on theoretical, experimental, and numerical fronts.This review aims to sort out the main complexities in OB: Oberbeck-Boussinesq BC: boundary condition turbulent RRBC and put the subject on a firm foundation.For that purpose, we confine our review to RRBC with Pr > 0.7 and leave out the very interesting topics of open-surface flows (e.g., Boubnov & Golitsyn 1986, Fernando et al. 1991, Bouillaut et al. 2021), convection in spherical geometry (Aurnou & Olson 2001, Busse 2002, Gastine et al. 2016, Guervilly & Cardin 2016, Kaplan et al. 2017, Long et al. 2020, Wang et al. 2021), and oscillatory convection for low-Pr fluids, applicable to liquid metals that constitute the Earth's outer core and the plasma in the convective zone of stars (e.g., see Chandrasekhar 1953, Zhang & Roberts 1997, Aurnou et al. 2018, Grannan et al. 2022).We also do not discuss the traveling wave modes of RRBC (Zhong et al. 1991, Ecke et al. 1992, Goldstein et al. 1993, Herrmann & Busse 1993, Kuo & Cross 1993) associated with the instability at Ra < Ra c , except with regard to their influence on the bulk convective state.Finally, we concentrate on results for intermediate Pr ≈ 7, owing to the large amount of data obtained in experiments using water.We also mention results for Pr ≈ 1 where geostrophic turbulence seems to be most accessible ( Julien et al. 2012b).
In this section we formulate the governing equations in the Oberbeck-Boussinesq (OB) approximation, introduce control and response parameters, and summarize the theoretical framework and the overall experimental-numerical characteristics of RRBC.Sections 2 and 3 focus on, respectively, the global flow structures and heat and momentum transport in different regimes of RRBC.In Section 4 we discuss RRBC configurations that deviate from the classical OB case, including the influence of strong centrifugation and different boundary conditions (BCs).We conclude with a discussion of open questions in RRBC and give an outlook on the experimental and numerical studies desired to better understand geostrophic turbulence in RRBC and in astro-and geophysical flows.

Governing Equations and Parameters
With a change of coordinate system from stationary to one rotating with angular velocity = e z , additional Coriolis (−2 × u) and centrifugal (− × × r) accelerations occur as additional effective body-force terms in the momentum equation: (for notational conventions, see the sidebar titled Dimensional Characteristics of RRBC).For ≡ e z one obtains − × × r = 2 re r .
In the simplest OB approximation that admits buoyancy (Oberbeck 1879, Boussinesq 1903), all fluid properties are assumed to be constant except for the density in the buoyancy term, where it is taken to be linearly dependent on temperature, ρ ≈ ρ 0 [1 − α(T − T 0 )].Introducing the reduced pressure, p ≡ P + ρ 0 gze z − (ρ 0 /2) 2 r 2 e r , from |α(T − T 0 )| ≤ α /2 â 1, which holds in the OB approximation, we obtain P/ρ ≈ (P/ρ 0 )[1 + α(T − T 0 )], as well as the momentum equation in the OB approximation (e.g., Becker et al. 2006), which together with the continuity equation, ∇ • u = 0, and the heat equation, form the governing equations in OB RRBC.One can see from the last two terms in Equation 2 that for RRBC there is gravitational buoyancy that acts in the vertical direction (as in nonrotating RBC)

DIMENSIONAL CHARACTERISTICS OF RRBC
Buoyancy timescale: τ ff ࣕ H/u ff Viscous timescale: τ ν ࣕ H 2 /ν Thermal timescale: τ κ ࣕ H 2 /κ Coriolis timescale: τ ≡ 1/(2 ) Centrifugal timescale: τ c ≡ 1/( √ α ) Inertial timescale: τ i ࣕ H/U, with U the reference velocity Velocity field: u ࣕ (u r , u φ , u z ), with radial e r , azimuthal e φ , and vertical e z components Convective free-fall velocity: u ff ≡ αg H Pressure: P, with p ≡ P + ρ 0 gze z − (ρ 0 /2) 2 r 2 e r the reduced pressure Temperature: T, with T = T + (T = T − ) at the bottom (top), ࣕ T + − T − , T 0 ࣕ (T + + T − )/2 Total vertical heat flux per unit area: q, with q 0 ࣕ C p κρ /H the heat flux from conduction Density: ρ, with ρ ≈ ρ 0 [1 − α(T − T 0 )] in the buoyancy term in the Oberbeck-Boussinesq approximation Isobaric thermal expansion coefficient: α Specific heat capacity: C p Kinematic viscosity of the fluid: ν Thermal diffusivity of the fluid: κ Angular velocity: ≡ e z , with the angular rotation rate Gravity vector: g ≡ −ge z , with g acceleration of gravity but also centrifugal buoyancy that acts in the radial direction and can only be neglected if is small.Note that in several textbooks, including that of Chandrasekhar (1961), the centrifugal term is evaluated under the assumption that the density is constant, which allows one to put the whole centrifugal term into the reduced pressure and thus leads to an equation similar to Equation 2, but without the last term.To study the centrifugal effects within the OB approximation, however, one needs to consider the full momentum Equation 2 (as in, e.g., Becker et al. 2006;Marques et al. 2007;Lopez & Marques 2009;Scheel et al. 2010;Horn & Aurnou 2018, 2019, 2021).The standard BCs for Equations 2 and 3 are no slip for the velocity (u = 0) at all walls, isothermal temperature at the bottom (T + ) and top (T − < T + ), and adiabatic (T/n = 0) at the sidewall.
Flow dynamics, global structures, and scaling relations of the heat and momentum transport in RRBC are determined by the dimensionless control parameters (see the sidebar titled Dimensionless Quantities in RRBC), which can be understood as ratios of involved forces or as ratios of related timescales.The dominance of one force over others determines transitions from one regime to another.These control parameters or suitable combinations occur explicitly in the corresponding dimensionless governing equations, which depend on the choice of the reference quantities.For example, taking as the reference quantities H/ αg H for time, for temperature, H for length, αg H for velocity, and ρ 0 αg H for the reduced pressure, one obtains dimensionless equations that look similar to Equations 2 and 3, but with the following substitutions for the viscosity ν → √ Pr/Ra, the thermal diffusivity κ → 1/ √ PrRa, the reduced pressure term p/ρ 0 → p, the Coriolis force term 2 e z × u → Ro −1 e z × u, the gravitational buoyancy term α(T − T 0 )g → T, and the centrifugal buoyancy term α(T − T 0 ) 2 re r → (2Fr/ ) T re r .Comparing the dimensionless gravitational and centrifugal buoyancy terms, one concludes that the latter, −α(T − T 0 ) 2 re r , is negligible for Fr 1.In the following, we focus on the case of Fr 1 but return to the influence of centrifugation in Section 4.

Theoretical Background, Main Features, and Scaling Properties
There are several perspectives that one may adopt for describing turbulent RRBC.The first is to treat the system in the rotation-dominated limit beginning at the onset of convection, where the Taylor-Proudman constraint and quasi-geostrophy play major roles, and to increase the effect of buoyancy until it becomes dominant (i.e., maintain constant Ek while increasing Ra).The second is to perturb the buoyancy-dominated state of turbulence by increasing rotation (decreasing Ek) at constant Ra.We discuss both here but take as a starting point the rotation-dominated limit because the major elements of RRBC-rotational suppression of convection, Taylor-Proudman constraint, Ekman boundary layer (BL), quasi-geostrophic convection, etc.-arise from that perspective.

Onset of bulk convection, nonhydrostatic quasi-geostrophic balance, and the
Taylor-Proudman constraint.Rotation postpones the onset of convection in the bulk of the domain.Linear stability analysis (Chandrasekhar 1953, Niiler & Bisshopp 1965, Homsy & Hudson 1971) shows that bulk convection sets in at a critical Ra = Ra c in the form of either steady or oscillatory flow, depending on Pr: where a reduction of 2 13/6 π 2/3 Ek 1/6 in the large-Pr case is a correction for no-slip BCs at the plates (see also Homsy & Hudson 1971).The onset of bulk convection in the form of a steady flow is preferred for 1 + Pr < 8Pr 4 (i.e., for Pr 0.6766 ≈ 0.68).
For rapid rotation (Ek → 0) there is approximate geostrophic balance between Coriolis and pressure-gradient terms in Equation 1 (Boubnov & Golitsyn 1986, Sakai 1997, Sprague et al. 2006, Julien et al. 2012b, King & Aurnou 2013, Aurnou et al. 2020, Aguirre Guzmán et al. 2021).Taking the curl of geostrophic balance, for an incompressible flow and = e z with a constant , one derives 0 ≈ ∇ × ( × u) ≈ −( • ∇)u and, hence, the result u/z = 0, which is known as the Taylor-Proudman constraint (Taylor 1921, Proudman 1916).In order to sustain convective vertical motion, this constraint must be broken in cells of finite vertical extent.Nevertheless, the constraint gives an intuitive sense of the suppression of the onset of convection and the strong anisotropy between lateral and vertical length scales and timescales, as quantified below.Owing to convection, the vertical force balance is nonhydrostatic and the lateral balance has smaller-order nongeostrophic contributions leading to a state of approximate or quasi-geostrophy.We denote the asymptotic description (Sprague et al. 2006, Julien et al. 2012b) of RRBC in the rapidly rotating limit as nonhydrostatic quasi-geostrophic.A recent numerical study of the force balance in RRBC (Aguirre Guzmán et al. 2021) quantifies the quasi-geostrophic balance for a considerable region above the onset of convection in both the interior and BLs, with nongeostrophic forces contributing at around the 10% level.Reflecting the high degree of vertical to horizontal anisotropy, the length scale in the vertical direction at the onset of convection is the height H, whereas the horizontal length is c (1/2 the critical wavelength), given by c H ≈ 2 1/6 π 2/3 Ek 1/3 ≈ 2.4Ek 1/3 , Pr 0.68, 2 1/6 π 2/3 (1 + Pr −1 ) 1/3 Ek 1/3 ≈ 2.4(1 + Pr −1 ) 1/3 Ek 1/3 , Pr 0.68.

5.
This is obtained from linear stability analysis for Cartesian (Chandrasekhar 1953)   (J n is the Bessel function of the first kind).For large ࣕ D/H â 1, one obtains (Shishkina 2021) the relations of Equations 4 and 5 in the limit Ek(1 + Pr)/Pr → 0. In the steady case we have ω 0 = 0, whereas in the oscillatory case, the oscillation frequency ω 0 in the same limit Ek/Pr → 0 follows

Main features.
An overview of the features of RRBC starting at onset can be seen in heat transport (Nu) measurements, covering almost 10 (5) decades in Ra (Ek) using water with Pr ≈ 7 (see Figure 1b).The rapid rise from the conduction value Nu = 1 represents the nonlinear growth from onset and defines a region of rotation-dominated quasi-geostrophic dynamics with intrinsic time dependence arising from nonlinear instability (Küppers & Lortz 1969, Cox & Matthews 2000) at any Ra > Ra c .As demonstrated by these data, the range of Nu spanned in this region increases with decreasing Ek so that in the limit Ek → 0, the buoyancy-dominated state becomes out of reach.In this limit, one can write a system of nonhydrostatic quasi-geostrophic equations valid asymptotically as Ek → 0, Ro → 0, and Ra → ∞ such that Ra ≡ Ra/Ra c ∼ Ek 4/3 Ra remains LSC: large-scale circulation finite (Sprague et al. 2006, Julien et al. 2012b).This approach holds promise to bridge to geo-and astrophysical systems with extreme values of Ra and Ek and provides a theoretical framework for describing the evolution of states in the rotation-dominated regime, often called the geostrophic regime, in which states of cellular flows, Taylor columns, plumes, and geostrophic turbulence are seen in experiments and solutions of the reduced equations (see Figure 1d-f ).
One common characteristic of the rotation-dominated regime is that thermal instability gives rise to vortical motions through the Coriolis force, −2 × u (i.e., horizontal flows are deflected by the Coriolis force, leading to vortices).The other important element of RRBC is the altered nature of BLs.The vorticity in the interior interacts with the no-slip plates through kinetic BL of the Ekman type (Greenspan 1968, Stellmach et al. 2014, Julien et al. 2016) and can generate Ekman pumping where the interior vorticity is dissipated in the Ekman BL of thickness δ E /H ∼ Ek 1/2 , and positive (negative) vertical velocity is produced for cyclonic (anticyclonic) vorticity.In the rotation-dominated regime, the kinetic BL (δ u ∼ δ E ) is much thinner than the thermal BL, which has a different structure from the nonrotating thermal BL, δ θ /H ≈ (2Nu) −1 , in which velocities are assumed to be small and heat is effectively carried diffusively.Owing to Ekman pumping in the rotation-dominated regime, advective processes in the thermal BL are significant so it is described as a thermal wind layer ( Julien et al. 2016) to differentiate it from the nonrotating thermal BL.An important aspect of this structure is that H/(2δ θ ) underestimates Nu owing to a combination of a finite mean gradient and contributions from Ekman pumping.
As Ra increases at constant Ek, the balance of rotation and buoyancy shifts, as measured by increasing Ro ∼ Ra 1/2 such that Nu approaches the nonrotating curve Nu 0 ∼ Ra 0.3 .For some ranges of Ro, Ra, and Pr 1, with δ u ≈ δ θ (King et al. 2009), Ekman pumping-amplified heat transport results in Nu ≥ Nu 0 (e.g., Rossby 1969) (see Figure 1b).For other parameter ranges (e.g., Pr 1 and Ra 10 10 ), Nu is always less than Nu 0 and asymptotes to its buoyancy-dominated dependence at some Ro t 1 that depends on various control parameters.Thus, we identify three regimes of RRBC: rotation dominated for Ro 1, buoyancy dominated for Ro 1, and rotation affected for Ro ∼ 1.The transition from the buoyancy-dominated through the rotation-affected and into the rotation-dominated zone is well represented by increasing rotation at fixed Ra.This approach, when normalizing by the nonrotating Nu 0 , also most readily demonstrates the enhancement of heat transport in the rotation-affected region for Pr 1 and is shown in Figure 1c.The different regions are indicated for this range of Ra ∼ 10 8 , although, as we will see, these boundaries depend on Ra, Pr, and as well.Characteristic experimental images in Figure 1 show rotation-dominated (Figure 1d-f ), rotation-affected (Figure 1g), and buoyancy-dominated (Figure 1h-i) states.

Thermal, Ekman, and Stewartson boundary layers and Ekman pumping.
The Nu behaviors in different regimes of RRBC are intimately related to BLs at the top and bottom of the convection cell.In the buoyancy-dominated state, the thermal BL thickness is given by δ θ ≈ H/(2Nu) and the kinetic BL is determined through the shear produced by the coherent accumulation of thermal plumes at the lateral boundaries into a large-scale circulation (LSC) (see Ahlers et al. 2009b).The resultant BL becomes thinner as the shear velocity increases with Ra.When the system rotates sufficiently rapidly, the nature of the kinetic BL changes dramatically to an Ekman BL type.An Ekman BL is formed when a fluid in solid body rotation at angular rotation rate experiences a small differential angular rotation ± ( â ) on a horizontal bounding surface (Greenspan 1968).In RRBC, the Ekman BL arises from the growth of thermal perturbations that lead to vertical and horizontal motion.For weak forcing with Ra ∼ 1, cellular circulation takes place with cyclonic outflow away from the plates and anticyclonic reversal at the midplane (Veronis 1959, Sakai 1997) (see Figure 2a).In the general case, at larger Ra and for noslip BCs, near the top (bottom) surface, the converging hot (cold) horizontal flow is acted upon by the Coriolis force to generate cyclonic vorticity in the rotating frame so that the local rotation rate of the fluid exceeds .Similarly, the return flow from the bottom (top) to top (bottom) (where it exists for Ra ∼ 1) spreads out as it comes near the top (bottom) and spins down anticyclonically so that the local rotation rate is less than .Thus, in both regions an Ekman BL forms with thickness δ u ∼ δ E to dissipate the interior flow vorticity at the no-slip horizontal boundary.Note that, as opposed to the buoyancy dominated/nonrotating kinetic BL, the thickness of the Ekman BL only depends on Ek and not on the strength of the velocity (i.e., on Ra).Near the onset of convection, we have δ u â δ θ so that Ekman pumping produces additional u z ∼ EkHω z .The vertical velocity amplification can be formulated as an effective Ekman pumping BC ( Julien et al. 2016) that yields a much steeper variation of Nu with Ra than one would expect for the rotation-dominated regime with no Ekman BL (Ek → 0 or free-slip BCs) ( Julien et al. 2012b, Stellmach et al. 2014, Plumley et al. 2016) and is consistent with measurements (Cheng et al. 2015, Lu et al. 2021) and direct numerical simulations (DNS) (Stellmach et al. 2014, Aguirre Guzmán et al. 2021).
Another effect of note (Kunnen et al. 2011) is that a mean volume-averaged anticyclonic vorticity is observed to form in the interior of the cell, which implies a net Ekman suction into the BL of order u z z=δ E ≈ (Ek/2) ω z H.A global circulation model (Kunnen et al. 2013) arises from these considerations where the flow into the BLs induced by the interior anticyclonic circulation balances a flow at the vertical boundaries in the form of Stewartson BLs (Stewartson 1957, Greenspan 1968) of thickness ∝ Ek 1/3 (Kunnen et al. 2011(Kunnen et al. , 2013)).This model explained the observed layer formation but did not address the origins of the flows.Recently, however, it was demonstrated that a boundary zonal flow (BZF) (Zhang et al. 2020(Zhang et al. , 2021;;de Wit et al. 2020;Wedi et al. 2021) arises over a wide range of Ra and Ek from the robustness of sidewall-traveling wall modes in the presence of bulk convection (Favier & Knobloch 2020, Ecke et al. 2022).Furthermore, the sidewall eigenfunctions for the vertical velocity of linear wall modes act mainly in the radial width that scales as ∝ Ek 1/3 (Herrmann & Busse 1993), in agreement with the results of Kunnen et al. (2011) and Zhang et al. (2020Zhang et al. ( , 2021)).Strong bulk turbulence modifies the BZF in ways still to be explored, but to first order the origin of the ∝ Ek 1/3 layer seems to result from the BZF as the source of vertical motion near the sidewalls with nonlinearities feeding back to produce the anticyclonic interior flow.The two descriptions seem to complement one another, with the Ekman-Stewartson mechanism feeding back on the wall mode/BZF state in a self-consistent manner.It is interesting that the wall mode vertical velocity profile, the Stewartson BL thickness, and the critical wavelength of bulk instability all scale as Ek 1/3 .The thermal and kinetic BLs at the plates interact to affect heat transport and other local properties of the flow.Prominent features include the finite mean temperature gradient in RRBC resulting from enhanced (decreased) lateral (vertical) mixing (e.g., Julien et al. 1996Julien et al. , 2012b;;Hart & Ohlsen 1999;Kunnen et al. 2009;Zhong et al. 2009;Stevens et al. 2010a;Liu & Ecke 2011;King et al. 2013;Horn & Shishkina 2014) and the character of statistical moments of T, u, and ω z .

Scaling properties.
Scaling relationships among measured quantities and control parameters depend on the location in RRBC parameter space and can be derived in certain limiting cases using approaches useful for nonrotating RBC.Under the assumption that the scaling relations for Nu ≡ qH/(κ ) have the form Nu ∼ Pr β 0 Ra γ in nonrotating/buoyancy-dominated regimes and Nu ∼ Pr β (Ra/Ra c ) ξ in rotation-dominated regimes (for certain γ > 0 and ξ > 0), along with the further assumption that the dimensional heat flow q is independent of diffusion in the BLs (i.e., of κ and ν), one immediately obtains for nonrotating/buoyancy-dominated regimes, 6.
Equations 6 and 8 for the buoyancy-dominated state are those of regime IV of the Grossmann & Lohse (2000) theory for nonrotating RBC (see also Kraichnan 1962, Spiegel 1971).Equations 7 and 9 for rotation-dominated RRBC are those of the geostrophic turbulence regime ( Julien et al. 2012b; also see Schmitz & Tilgner 2009).Equating the scaling relations for Nu (or Re) in the buoyancy-dominated and rotation-dominated regimes, one obtains that the scaling quantity is Ro.Thus, if the above assumptions are fulfilled, the transition from the buoyancy-dominated to rotation-dominated geostrophic turbulence regimes scales with Ro, Nu/Nu 0 ∼ Ro 2 , and Re/Re 0 ∼ Ro in the rotation-dominated regime and Nu/Nu 0 ∼ 1 and Re/Re 0 ∼ 1 in the buoyancydominated regime.The same scalings of Equations 6-9 also follow from Aurnou et al.'s (2020) approach, where relevant scales for length , velocity U, and temperature θ were introduced; the scalings Nu ∼ θU/(κ /H ) and Re ∼ U H/ν were assumed; and terms in the vorticity equation were analyzed.For the rotation-dominated regime, Aurnou et al. (2020) proposed a balance of the Coriolis term (estimated as 2 U/H), the inertial term (U 2 / 2 ), and the buoyancy (Archimedean) term (gαθ/ ), which together with θ/ ∼ /H lead to U/u ff ∼ Ro, /H ∼ Ro and to Equations 7 and 9 for the rotation-dominated state.For the buoyancy-dominated regime, the inertia buoyancy balances together with θ ∼ and ∼ H, and U ∼ u ff leads to the scalings of Equations 6 and 8.
In other parameter ranges, different assumptions can be taken to derive scalings.Thus, assuming that q is independent not of κ and ν but of H, one derives Nu ∼ Ra 1/3 for the buoyancydominated regime (Malkus 1954, Priestley 1959) and Nu ∼ (Ra/Ra c ) 3 for the rotation-dominated state.No experimental or numerical (under realistic conditions) evidence exists, however, to support a scaling range of this latter type (also see Julien et al. 2012b).Finally we mention that a rigorous upper bound on the heat transport in RRBC derived by Grooms & Whitehead (2014), Nu ≤ 20.56 Ek 4 Ra 3 , is obtained from the asymptotically reduced equations for Ek 8/5 Ra = O(1) in the limit of rapid rotation (Ek → 0), strong thermal forcing (Ra → ∞), and infinite Prandtl number.For free-slip BCs and infinite Pr, Tilgner (2022) suggests the upper bound Nu < 0.144 Ek 2/3 Ra.Note that the existing theoretical upper bounds are much larger than the measured ones.In Section 3 we check several hypotheses and suggest a unifying theoretical scaling model for the transition between rotation-dominated and buoyancy-dominated regimes.

Experimental and Numerical Investigations of RRBC
RRBC experiments (see examples in Figure 3) and DNS aim to capture a broad parameter range in the Ra-Ek plane in order to investigate the many regimes of RRBC.These approaches are highly complementary with DNS, as they can access the full hydrodynamic fields and have the flexibility to investigate different BCs and explore combinations of control parameters that are inaccessible experimentally.Experiments, on the other hand, can easily accumulate statistical averages over much longer times, reflect realistic limitations on idealized descriptions, and access a different range of control parameters.Of particular interest is the rotation-dominated regime of very high Ra and very small Ek, which reflects the nature of astrophysical and geophysical flows.
For a given container and a particular fluid, the range of experimental values of Ek is restricted by the possible rotation rates , and the Ra range is restricted by the maximal and minimal temperature differences that can be imposed between the bottom and top plates.The requirement to satisfy OB conditions puts additional restrictions on the imposed and on the range of Ra (Gray & Giorgini 1976, Horn & Shishkina 2014, Weiss et al. 2018) (see Figure 4a and Section 4 below).The other two demanding requirements are minimizing centrifugal effects (Fr 1) and limiting the impact of vertical sidewalls so that the horizontal extent of typical structures in RRBC is much smaller than the cell diameter D (the recently discovered BZF complicates this latter criterion).Therefore, the available range is bounded from above by centrifugal effects (Fr ∝ 2 ) and from below by the cell aspect ratio because of ∝ −1/3    Marques 2009, Cheng et al. 2018, Horn & Aurnou 2018).The resulting parameter range in a realistic OB experiment is bounded for both Ra and Ek (see Figure 4b).
Another method to characterize the structures and properties of the rotation-dominated regime is to asymptotically reduce the full OB RRBC equations in the limit Ek → 0 (Ro → 0) and Ra → ∞ while keeping laterally periodic BCs and finite Ek 4/3 Ra (Nieves et al. 2014, Julien et al. 2016, Plumley et al. 2016, Plumley & Julien 2019).This results in nonhydrostatic quasi-geostrophic model equations that can be solved numerically.

FLOW STRUCTURES IN RRBC
Fluid flow organizes itself in RRBC in rich and diverse ways.We begin our discussion with the rotation-dominated regime and systematically vary control parameters to access domains with decreasing rotational influence.Taking Ek fixed (as in Figure 1b) and increasing Ra, one sequentially finds wall modes for Ek −1 Ra Ek −4/3 ; nonhydrostatic quasi-geostrophic features of RRBC including cellular flows, convective Taylor columns, plumes, geostrophic turbulence, and large-scale vortex condensates in the range 1 ≤ Ra 50 to 100; the transition to the rotation-affected region with remnant spatial anisotropy and Ekman BL effects; and finally the buoyancy-dominated domain.Representative images from experiment, DNS, and the nonhydrostatic quasi-geostrophic model for each region are shown in Figure 1d-g and Figure 5.
The first instability to a convecting state in rotation-dominated convection is to wall modes (Zhong et al. 1991, Ecke et al. 1992, Goldstein et al. 1993, Herrmann & Busse 1993, Kuo & Cross 1993), which are confined near the sidewall, have azimuthal periodicity m, precess in a retrograde direction, and have onset at Ra wm ≈ π 2 6 √ 3Ek −1 .These states are important for considering how the system enters the quasi-geostrophic regime and undergoes a transition to turbulence owing to the surprising robustness of the wall modes as a BZF that appears to persist over the entire rotation-dominated and rotation-affected regimes (Favier & Knobloch 2020;Shishkina 2020;de Wit et al. 2020;Zhang et al. 2020Zhang et al. , 2021;;Wedi et al. 2021;Ecke et al. 2022).These states are particularly influential for small , where they contribute substantially to the total heat transport.

Quasi-Geostrophic Convection
Several states of RRBC have been identified using the nonhydrostatic quasi-geostrophic approach (Sprague et al. 2006, Julien et al. 2012b).Near onset, RRBC takes the form of cellular vortical structures (Chandrasekhar 1953, Veronis 1959) (see Equations 4 and 5 and Figure 2a).These structures (Figure 5a,c,d) are nonlinearly unstable to slow dynamics (Küppers & Lortz 1969, Cox & Matthews 2000).With increasing Ra and Pr 3, the flow structures gradually change to convective Taylor columns (Figure 1d and Figure 5e) with an interesting structure of T and ω z (Grooms et al. 2010, Rajaei et al. 2017) around the Taylor column and efficient heat transport.With further increase of Ra, the vertical coherence of the convective Taylor columns is degraded and plumes only partially penetrate across the fluid layer with an even shorter vertical correlation length ( Julien et al. 2012b, Nieves et al. 2014, Rajaei et al. 2017) (see Figure 5f,g).For the highest investigated Ra ≈ 20 in the nonhydrostatic quasi-geostrophic model ( Julien et al. 2012b) and for Pr = 1 (see Figure 5g), one reaches a state of geostrophic turbulence in which heat transport is throttled by the interior rather than by thermal BLs.Experimental examples of geostrophic turbulence are shown in Figure 5h with Ra = 77 and Ro c = 0.12 and in Figure 1f with Ra = 27 and Ro c = 0.024.Finally, in certain circumstances DNS have shown that a large-scale vortex condensate forms from the geostrophic turbulence state ( Julien et al. 2012b, Guervilly et al. 2014, Guervilly & Hughes 2017, Julien et al. 2018, Favier et al. 2019, de Wit et al. 2022).The regions of geostrophic turbulence and large-scale vortex condensate have sufficiently large Ra such that for experiments and DNS one may no longer be in the quasi-geostrophic regime but rather in the rotation-affected regime; large-scale vortex states have not been observed in experiments.Eventually, with increasing Ra the flow enters the buoyancy-dominated regime, where the flow closely resembles turbulent nonrotating convection (Nieves et al. 2014, Cheng et al. 2015, Julien et al. 2016, Cheng et al. 2018) (see Figure 1h,i).A detailed review of the quasi-geostrophic regime and its different regimes has been given by Kunnen (2021).Although the qualitative and semiquantitative features predicted from the quasi-geostrophic model agree fairly well with observations from experiments and DNS, significant details remain to be determined, including how Ekman pumping and non-quasi-geostrophic effects manifest in systems at finite Ek and Ro.As a starting point, the solutions of the nonhydrostatic quasi-geostrophic equations exhibit for Pr = 7 transitions from cellular to convective Taylor columns state at Ra ≈ 2, transitions from convective Taylor columns to plumes at Ra ≈ 6, and plumes to geostrophic turbulence at Ra > 17 (no transition is observed).5a-c) is characterized by quasi-steady convective Taylor columns (Veronis 1959, Heard & Veronis 1971, Sakai 1997) that are built of vertically coherent plumes emitted synchronously from the hot and cold BLs.Each column consists of a hot (cold) central region with a cyclonic (anticyclonic) vortex core surrounded by a region of opposite temperature contrast and oppositely signed vorticity (Sprague et al. 2006;Grooms et al. 2010;Julien et al. 2012b;King et al. 2012;Rajaei et al. 2016Rajaei et al. , 2017)).The convective Taylor columns are nonuniformly distributed horizontally and the vortices undergo complex interactions including vortex mergers (Zhong et al. 1993, Noto et al. 2019), a flux of vortices away from the lateral boundary (Noto et al. 2019, Ding et al. 2021) attributed to centrifugal effects, and a diffusive-like motion of individual vortices in the overall vortex array (Chong et al. 2020, Ding et al. 2021).In this region, the shielding of the convective Taylor columns reduces vortex interactions and stabilizes the vortex array (Grooms et al. 2010, Rajaei et al. 2017).For experimentally and computationally accessible Ek > 10 −8 , the effects of Ekman pumping are large (Stellmach et al. 2014, Julien et al. 2016, Plumley et al. 2016) and lead to a very rapid increase in Nu in this regime for Pr ≈ 7 compared to the expected asymptotic linear dependence, Nu − 1 = a + O( 2 ), with a ≈ 2 (Bassom & Zhang 1994, Dawes 2001).Figure 6a shows Nu − 1 versus ϵ (reflecting Nu = 1 at Ra = 1) for a variety of data (after figure 5 of Stellmach et al. 2014).The lowest values of Ra are consistent with the weakly nonlinear solution with a ≈ 2, but the O( 2 ) term is 20 times larger than the nonhydrostatic quasi-geostrophic results, indicating that Ekman pumping is felt even very close to onset.The power law relationship for the data in the convective Taylor columns regime is Nu − 1 ∼ ( Ra − 1) 5/3 , as compared to Nu ∼ Ra 3 (Stellmach et al. 2014);

Columnar regime (for Pr 3). The columnar regime (Figure
the effective scaling exponent is very sensitive to subtracting 1 unless Nu 1 and Ra 1.One also notes from Figure 6a that the convective Taylor columns persist to higher Ra for smaller Ek.For Pr = 1, Ekman pumping has a similar effect on the slope of Nu near onset, although the convective Taylor columns regime is not observed ( Julien et al. 2012b).Other characterizations of the convective Taylor columns state include a rapid decrease in the mean temperature gradient (Figure 6b) and increasing normalized root-mean-square (rms) averages ω zrms (Figure 6c) and T rms (Figure 6e) for the nonhydrostatic quasi-geostrophic model.Available experimental data (Vorobieff & Ecke 2002, Shi et al. 2020) agree with the increase of (ω zrms / )Ek −1/3 in this regime.Although there is no experimental data in this regime for T rms , DNS data (Aguirre Guzmán et al. 2022), when normalized as (T rms / )Ek −1/3 , show a similar increasing trend for small Ra with a similar magnitude.QG approach (Julien et. al. 2012b) QG approach (Julien et. al. 2012b) (d) rms of the temperature normalized by its maximum value T rms /(max z T rms ) versus z/δ θ , the distance from the plate normalized by the thermal BL thickness.Arrows indicate the approximate locations of the normalized thicknesses of the kinetic BL δ u /δ θ for the nonhydrostatic quasi-geostrophic (δ < u ) and power law (δ > u ) scaling regions for larger Ra (Aguirre Guzmán et al. 2021).(e) rms of the temperature normalized with , the temperature difference between the plates, (T rms / ) Ek −1/3 [or (T rms / ) Ro −1/3 ] versus Ra. ( f ) Vortex densities λ 2 c N + and λ 2 c N − versus Ra: cyclonic (blue) and anticyclonic (red).For all panels, the Prandtl number is Pr ≈ 7. Abbreviations: BL, boundary layer; C, cellular; CTC, convective Taylor columns; DNS, direct numerical simulations; exp., experiment; GT, geostrophic turbulence; LSV, large-scale vortices; P, plumes; QG, quasi-geostrophy; rms, root-mean-square.

Plume regime.
The convective Taylor columns lose vertical coherence in the plume regime for Pr 3 (Figure 5f ) so that individual vortical structures terminate in the interior.In losing their coherence (King & Aurnou 2012), the plumes become increasingly desynchronized across the layer at larger Ra (Sprague et al. 2006, Julien et al. 2012b, King & Aurnou 2012, Rajaei et al. 2017).Nieves et al. (2014) and Cheng et al. (2020) suggested that a transition from columnar convection to a plume regime for large Pr (Pr ≈ 7) takes place at about Ra ≈ 6.This identification is consistent with DNS and experiments for 10 −8 Ek 10 −4 in terms of the following features: a change in slope of Nu versus Ra, a saturation of T/z, a reversal in the slope of T rms (see Section 2.1.3),and a trend toward equalization of cyclonic and anticyclonic vortex density, as illustrated in Figure 6a,b,e, and f, respectively.

Geostrophic turbulence.
The nonhydrostatic quasi-geostrophic model ( Julien et al. 2012b) predicts that at Ra ≈ 5Ra c (for Pr 3), there is a transition to geostrophic turbulence where the geostrophic balance is maintained with the interior of the flow, as well as in the BL.In this regime, the vortical structures are very short and attached to the BLs, whereas the bulk flow is well mixed laterally and turbulent.Owing to geostrophic balance, the flow structures maintain a degree of vertical alignment (Figure 1f ) and an interior control of heat transport, as opposed to a BL-controlled process.The boundary separating plumes and geostrophic turbulence is more complex than the transition from convective Taylor columns to plumes (Cheng et al. 2020).For Pr = 7, geostrophic turbulence was not found in the nonhydrostatic quasi-geostrophic simulations.In Figure 6b. the structure of the finite mean temperature gradient shows a saturated region in the range 10 Ra 300 corresponding to 10 8 ≤ Ra ≤ 10 9 , which decreases in range as Ra increases or Ek decreases (the Pr factor multiplying −T/z yields a better collapse of the data, with lower saturated mean gradient increasing with increasing Ra).The rapid decrease in slope for Ra ≈ 300 corresponds to Ro ≈ 1.For the largest 10 11 Ra 10 12 , there is a Ra −1/2 scaling over about 1.5 decades in Ra and a maximum at the slightly smaller Ra ≈ 50 [a shallower slope was fit (Cheng et al. 2020) to a subsection of the data close to the maximum with an effective slope of −0.21, rather than the −1/2 identified here, which is consistent with Hart & Ohlsen (1999)].For this Ra range, Ra is greater than 5,000 for Ro ≈ 1.The nature of the saturation, the maximum, and the power law decrease are not fully understood; perhaps geostrophic turbulence modified by ageostrophic contributions exists for the lower Ra range of these observations, although the Nu scaling is not consistent in this range with the nonhydrostatic quasi-geostrophic prediction of Ra 3/2 (Figure 6a).There is a striking difference in the trends of T rms evaluated at its maximum value (with respect to z) versus Ra (Figure 6e).The nonhydrostatic quasi-geostrophic model suggests normalizing T rms with Ek −1/3 , and the experimental results are similar in magnitude when normalized in this manner, although the trends with Ra are the opposite.DNS data (Aguirre Guzmán et al. 2022) for 5 × 10 9 ≤ Ra ≤ 1.5 × 10 12 span the range 1.3 < Ra < 80 and bridge the gap between nonhydrostatic quasi-geostrophic predictions and experiment.The lower Ra values agree well with the nonhydrostatic quasi-geostrophic results, whereas the Ra = 80 value is significantly (by about a factor of 3) larger than the corresponding experimental data.For nonhydrostatic quasi-geostrophic results, one has (T rms / )Ek −1/3 ∼ Ra 7/4 for Ra ≤ 7 approaching a constant for higher values, whereas the data from experiments with 3 × 10 8 Ra 5 × 10 9 show scaling to be approximately ∼ Ra −2/5 up to about Ra ≈ 500.A scaling that better collapses the DNS and experimental data is (T rms / )Ro −1/3 , implying that there is a remnant Ra (and Ek) dependence that accounts for the magnitude shift.The maximum of T rms at Ra ≈ 7 indicates that the scaling for larger Ra ≤ 7 results from the plume transition and is not affected by the crossover to geostrophic turbulence.A final feature of T rms , for δ u ≥ δ θ , is that its variation with z takes on a universal shape of a maximum value at the thermal BL thickness with different effective power law scalings inside and outside that layer (Ding et al. 2019, figure 6d).For z/δ θ < 1, we have T rms (z) ∼ z 0.9 , independent of rotation, whereas outside the BL one has an Ro-dependent effective scaling going from ∼z −0.6 for the nonrotating case to ∼z −0.25 , where the effective scaling exponent is approximately constant for Ro 0.5 (Ding et al. 2019, figure 2b).For these data, one has δ θ ≈ δ u , indicating that they are not in the quasi-geostrophic limit.The nonhydrostatic quasi-geostrophic results are quite different for δ θ â δ u , with similar magnitude but without an effective power law scaling for z/δ θ > 1, but with much larger fluctuations and again no power law scaling for z/δ θ < 1, where presumably Ekman pumping effects play a leading role (Stellmach et al. 2014, Julien et al. 2016).
Geostrophic turbulence can support an inverse energy transfer process that results in the formation of large-scale vorticity.Although similar in outcome (i.e., a large-scale vortex condensate), the energy transfer is a direct one from small scales to large scales via 3D modes and thus is different from a purely 2D inverse energy cascade (Boffetta & Ecke 2012).These flows have been seen in DNS ( Julien et al. 2012b;Favier et al. 2014;Guervilly et al. 2014;Rubio et al. 2014;Stellmach et al. 2014;Kunnen et al. 2016;Kunnen 2021) for a variety of (Guervilly & Hughes 2017, Julien et al. 2018), for different Pr (Maffei et al. 2021), and for free-slip and no-slip BCs at the plates (Aguirre Guzmán et al. 2020).Such large-scale vortices, however, have only been observed in simulations with periodic BCs in horizontal directions, and the formation of these vortices depends on (Guervilly & Hughes 2017, Julien et al. 2018) and on initial conditions (Favier et al. 2019).Whether these large-scale vortex states persist in real experiments with sidewalls and with significant Ra 1, where other physics may dominate, remains an open question.
2.1.4.Boundary zonal flow and transitions to the rotation-affected regime.RRBC turbulence in a laterally confined geometry is also characterized by the presence of a BZF located close to the container sidewall (de Wit et al. 2020;Zhang et al. 2020Zhang et al. , 2021;;Wedi et al. 2021), which can be understood as the remnant of wall modes (Favier & Knobloch 2020, Ecke et al. 2022).
The BZF has been directly and indirectly observed from the onset of bulk convection up to the transition to buoyancy-dominated flows at Ro ≈ 2. It is especially influential in systems with ≤ 1, which have been utilized for experimental purposes to reach extremes of system parameters.The fluid motion within the BZF is cyclonic (prograde) whereas the temperature pattern drifts anticyclonically (retrograde).Within the BZF the vertical heat flux can be 60% larger than the average Nu, making the BZF particularly significant (Zhang et al. 2021).DNS by Ecke et al. (2022) for a fixed Ek = 10 −6 and varying Ra showed a direct connection between pure wall modes, which occur prior to bulk convection, and the BZF that coexists with turbulent convective flow in the bulk.The impact of the BZF on flows in realistic physical geometries is only recently beginning to be appreciated, and its existence may offer explanations for several observations, such as the varying slope of Nu versus Ra near onset [DNS by Lu et al. (2021)] and the attribution of a deficit of vortex density at the walls to centrifugal (Noto et al. 2019) or proximity (Weiss & Ahlers 2011b) effects.

Rotation-Affected and Buoyancy-Dominated Turbulence
Here we detail the transition from the rotation-dominated regime to the rotation-affected regime and the transition to buoyancy-dominated turbulence.The characteristics of these transitions are a change in slope of dNu/dRa and the parameters for which Nu → Nu 0 .For fixed Pr, one takes the quantity Ek α 0 Ra as the combined transition parameter, where α 0 is chosen to provide the best collapse of the data-one typically uses Nu as the indicator of the transitions.Theory and experiment provide several choices in the range 4/3 ≤ α 0 ≤ 2, with the limits corresponding to Ra ∼ Ek 4/3 Ra and Ro 2 ∼ Ek 2 Ra.These transitions are discussed in detail by Kunnen (2021).

Rotation-affected regime.
As seen in Figure 1b, Nu at constant Ek increases rapidly with Ra from its unity value at onset toward an asymptotic value corresponding to its nonrotating value Nu 0 .For Ra 10 10 and Pr 1, Nu exceeds Nu 0 owing to Ekman pumping, which gives a concrete transition point, Nu(Ra t , Ek) = Nu 0 .For larger Ra and smaller Ek, we have Nu ≤ Nu 0 , so that the transition value is given by a sharp decrease in the slope dNu/dRa.This rotation-affected regime, which separates the rotation-dominated and buoyancy-dominated regimes, is characterized by the presence of vortical thermal plume emission from the BLs and the absence of an LSC (Vorobieff & Ecke 2002;Kunnen et al. 2008;Weiss & Ahlers 2011a,b,c).The heat transport can be more efficient than it is for nonrotating convection owing to Ekman pumping (e.g., Rossby 1969;Zhong et al. 1993;Julien et al. 1996;Liu & Ecke 1997, 2009;Zhong et al. 2009;Stevens et al. 2010a;King et al. 2009) for Pr 1 and, if Ra is sufficiently small, Ra 10 10 .King et al. (2009) suggested that the transition value into the rotation-affected regime for Pr ≈ 7 was determined by an approximate equivalence of thermal and Ekman BL thicknesses [i.e., δ θ ≈ (2Nu) −1 ≈ δ E with an empirical transition Ra t ≈ 1.4Ek −7/4 , which was later modified to Ra t ≈ 10Ek −3/2 (see King et al. 2012King et al. , 2013;;King & Aurnou 2013)].Liu & Ecke (2009) suggested Ro = 0.1, corresponding to Ra t ≈ 0.06Ek −2 .The nonhydrostatic quasi-geostrophic approach predicts a breakdown of geostrophic balance at Ra t ∼ Pr 3/5 Ek −8/5 .Measurements for Pr ≈ 0.7 (Ecke & Niemela 2014) gave Ra t ≈ 0.25Ek −1.8 and recast the results of Liu & Ecke (2009) as Ra t ≈ 1.3Ek −1.65 , in close agreement with results of King et al. (2009).Recent results (Lu et al. 2021) for Pr ≈ 4 indicated Ra t ≈ 0.2Ek 1.7 , while Wedi et al. (2021) found Ra t ≈ 0.8Ek −2 .Finally, Cheng et al. (2018Cheng et al. ( , 2020) ) identified a transition to a so-called unbalanced BL regime, which is associated with a breakdown of quasi-geostrophy in the thermal BL, while the quasi-geostrophic condition is maintained in the interior at Ro t ≈ 0.06.How exactly factors such as aspect ratio , Pr, or the contributions of a BZF in finite containers affect this transition has yet to be worked out in detail.Nevertheless, the rotation-affected regime is rich with interesting crossovers from the quasi-geostrophic to buoyancy-dominated flow.

From large-scale circulation to boundary zonal flow.
When rotation is slow, the turbulent RBC flow looks similar to the nonrotating case.In nonrotating turbulent RBC in containers of ∼ 1, an LSC develops (Ahlers et al. 2009b).There is a vertical central cross section (an LSC plane), in which a large LSC roll fills the core part with secondary rolls in the corners.The LSC can undergo twisting, sloshing, and other motions (see, e.g., Cioni et al. 1997;Funfschilling & Ahlers 2004;Xi et al. 2004Xi et al. , 2006;;Wagner et al. 2012).Since the LSC is always tilted, in another vertical cross section orthogonal to the LSC plane, the flow typically looks like a four-roll structure, with an inflow in the central horizontal cross section (Shishkina et al. 2013(Shishkina et al. , 2014)).Under slow rotation, at mid-height, the flow toward the center is affected by the Coriolis force in such a way that a cyclonic (prograde) fluid motion is induced there (Kunnen et al. 2011).Near the plates, however, the LSC flow toward the sidewall under the action of the Coriolis force leads to an anticyclonic (retrograde) fluid motion there (see Zhang et al. 2021, figure 2).Near the centerline and close to the plates, the flow remains anticyclonic for all .As rotation increases, the mean flow structure tends to be homogeneous in the vertical direction (according to the Taylor-Proudman constraint), which results in the following: The region of the anticyclonic motion grows from the plates toward the bulk and finally occupies the core part of the domain, whereas the region of cyclonic motion, which forms the BZF, is pushed toward the sidewall and shrinks with increasing (Ro −1 ).Thus, at a certain constant Ro of order one, a breakdown of the LSC happens (Vorobieff & Ecke 2002;Kunnen et al. 2008;Weiss & Ahlers 2011a,b,c) and a BZF becomes the most prominent system-spanning global structure (de Wit et al. 2020;Zhang et al. 2020Zhang et al. , 2021;;Ecke et al. 2022).

Other Characteristics of RRBC Flows
There are many other characteristics of RRBC flows that have been either computed from DNS or measured directly in physical experiment.Our review has selected what we view as the central quantities that represent the problem.Here we discuss several others of interest.

Toroidal and poloidal energies.
The change of flow structures is also tracked in the evolution of the toroidal (e t ) and poloidal (e p ) kinetic energies (Horn & Shishkina 2015).For moderate Ra, e t is less than e p if buoyancy dominates and e p gradually decreases with stronger rotation.With increasing Ro −1 , e t increases in the rotation-affected regime until it achieves e t = e p ; this location can be interpreted as the beginning of rotation dominance where e t is greater than e p .

Statistical moments of temperature, velocity, and vorticity.
The statistical moments of the convective fields (T, u, and ω z ) are important indicators of the physics of RRBC and elucidate its similarities and differences with nonrotating RBC ( Julien et al. 1996( Julien et al. , 2012b)).In experiments, single point probes of T are utilized (Liu & Ecke 1997, 2011;Hart & Ohlsen 1999;Hart et al. 2002;Ding et al. 2019) and the dependence on the distance to the plate z can be determined.Similar measurements can be performed using DNS (Kunnen et al. 2010a, Aguirre Guzmán et al. 2022).There is a maximum of T rms at z = δ θ and larger fluctuations at z = δ θ , as well as at the mid-plane z = H/2, compared to the nonrotating case.Both the skewness (sign preference) and kurtosis (Gaussianity measure) of T are smaller than they are for nonrotating RBC at a particular z, and they are small near the thermal BL and increase with z, reaching a max value at about 10δ θ (Liu & Ecke 2011, Ding et al. 2019); the skewness is approximately 0 at the mid-plane owing to symmetry [cf. Aguirre Guzmán et al. (2022) for the Taylor columns state].In DNS (Aguirre Guzmán et al. 2022), it is possible to evaluate statistics and average over horizontal planes, which are very difficult to obtain in experiments.The DNS show that the skewness is preferentially negative for δ u < δ θ but becomes positive for the reverse.This is consistent with single point measurements (Liu & Ecke 2011, Ding et al. 2019) inside the thermal BL for the conditions δ θ ≈ δ u .The kurtosis of T is uniformly greater than 3, indicating stretched exponential probability density functions (PDFs).The skewness of horizontal velocity and ω z are ≈0 (Vorobieff & Ecke 2002, Kunnen et al. 2010b, Aguirre Guzmán et al. 2022) at the mid-plane, with exponential tails rather than the Gaussian PDFs for nonrotating RBC.Near the plates there is skewness toward cyclonic vorticity and positive u z with strong non-Gaussianity.This is consistent with the picture in Figure 2 regarding cyclonic vorticity generation and Ekman pumping-induced upward flow, u z > 0. The nature of fluctuations near the plates is an important subject that needs additional study to further elucidate the interactions of the thermal and kinetic BLs in the presence of Ekman pumping (Stellmach et al. 2014, Julien et al. 2016, Plumley et al. 2016).

HEAT TRANSPORT IN RRBC
The global measure of RRBC is the convective enhancement of heat transport Nu.There are clear scaling relationships in the limits of rotation-dominated and buoyancy-dominated regimes, as discussed above.In between, the situation is less clear cut.We assume for the sake of comparison that there is power law scaling in the range of convective Taylor columns/plumes 2 Ra 10 (see Figure 6a), although Ekman pumping contributes significantly here.There are no theoretical predictions for this region.

Measurements and Simulations of Heat Transport in RRBC
Experimental measurements of Nu require fine system control and precision in heat flow [e.g., accurately measuring the heat flow through the fluid as opposed to measuring through sidewalls, evaluating non-OB effects, introducing proper heat shielding, and measuring control variables such as temperature difference and heat current (Ahlers et al. 2009b)].DNS measures complement experiments by allowing access to the full fields of temperature and velocity but require very fine grids and long computing times to obtain good statistical averages.To properly interpret Nu one must be aware of wall modes that contribute at the onset of bulk convection and continue to influence total Nu through the coexisting BZF.This contribution can be fractionally large for convection in small (e.g., Zhang et al. 2020, de Wit et al. 2020, Ecke et al. 2022).We now describe the evolution of Nu in different regimes, starting from onset Ra = 1 and ending with Ro 1 (i.e., the nonrotating limit).
After a small region of weakly nonlinear growth for ϵ 1, Nu rises steeply owing to Ekman pumping (Stellmach et al. 2014, Julien et al. 2016, Plumley et al. 2016) for Pr = 1 and Pr = 7, as Nu − 1 scales as ϵ 5/3 -or, without the subtracted value of 1, as Nu ∼ Ra 3 [earlier analysis and explanations did not include the important Ekman pumping effect (Boubnov & Golitsyn 1995;Canuto & Dubovikov 1998;King et al. 2009King et al. , 2012;;Ecke 2015)].The initial steepening increases with decreasing Ek (Cheng et al. 2015), but the modified nonhydrostatic quasi-geostrophic model suggests that Ekman pumping reaches its maximum at Ra ≈ 3 for Ek = 10 −7 (see Plumley et al. 2016, figures 4 and 14), with factors of 2.5 and 10 of higher Nu compared to the nonhydrostatic quasi-geostrophic prediction for Pr = 1 and Pr = 7, respectively.The latter is in the same range as the data presented in Figure 6a, where the maximum amplification factor is about 6.The region where the maximum enhancement occurs is within the convective Taylor columns or plume regime, depending on Pr.Following the rapid increase, the effective scaling of Nu with Ra continuously decreases toward an asymptotic buoyancy-dominated effective scaling of Nu ∼ Ra 0.3 (Cheng et al. 2020), although less slowly for smaller Ek (see Figure 6a).The same trend is seen in figure 3 of Lu et al. (2021), with the observation that there is an apparent wall mode/BZF contribution Nu ≈ 10, as per Ecke et al. (2022).How much of Nu in this region is contributed by the BZF is a matter of current research.In contrast, for Ek 10 −6 , Nu undergoes a rapid transition to BL-controlled convection with Ekman pumping enhancement with approximate Ra 1/3 scaling for 10 Ra 100.For smaller Ek, the transition is to ∼ Ra 1/2 and ∼ Ra 5/8 for Ek ≈ 10 −7 and 10 −8 , respectively.This may indicate a transition toward geostrophic turbulence with an expected Ekman pumping-adjusted dependence of Nu ≈ 0.04(1 + 5.97Ek 1/8 )Pr −1/2 Ek 2 Ra 3/2 ( Julien et al. 2012b, Plumley et al. 2017) for smaller Ek.
As Ra increases further, one arrives in a region where Ro is no longer small.The condition Ro ≈ 1 then yields an upper bound on the rotation-affected regime of Ra t ≈ (Pr/A)Ek −2/3 , where we have A = Ra c Ek 4/3 .For Pr = 7, one has 2,000 Ra t 2 × 10 5 for 10 −8 Ek 10 −5 .Based on the earlier quasi-geostrophic analysis, one can estimate that the transition from the quasigeostrophic to rotation-affected regime is in the range 100 Ra t 2 × 10 5 , depending on Ek.Within this range, different scalings and data-collapse strategies have been proposed.The effective scaling exponents are very sensitive to the Ra and Ek ranges where the fits are made, as this is exactly the region that connects very different scaling regimes of the dominance of rotation and of buoyancy.In the rotation-affected regime, for Pr 1 and not too large Ra 10 10 , Nu is larger than it is for the nonrotating case (Nu 0 ), owing to the positive Ekman pumping effect noted above and the relatively low value of Nu 0 (Stevens et al. 2010b(Stevens et al. , 2013;;Horn & Shishkina 2014).Based on DNS for 4.38 ≤ Pr ≤ 100 and 10 7 ≤ Ra ≤ 10 9 , Yang et al. (2020) obtained the optimal Ro −1 o ≈ 0.12 Pr 1/2 Ra 1/6 , at which, for fixed Ra and varying Ro, the maximal Nu/Nu 0 was observed.The optimal heat transport relative to Nu 0 was obtained for this range of Ra when the thicknesses of the thermal BL (estimated as ∼Ra −1/3 ) and the viscous Ekman BL (∼ Ek 1/2 ) are similar.For comparison with data in Figure 6a, one also has the result for optimal Nu at Ra ≈ 3Ek −1/6 , which yields something in the range of 10 Ra 20 for 8 × 10 −6 Ek 2 × 10 −4 .As one can see from Figure 6a, this brackets a quite narrow range in the nonhydrostatic quasi-geostrophic diagram and severely limits access to quasi-geostrophic states.
The transitions between different regimes may depend on .For example, a rotation-affected to buoyancy-dominated transition scales empirically as Ro −1 = c 1 −1 (1 + c 2 −1 ) with c 1 ≈ 0.38 and c 2 ≈ 0.06 (Weiss & Ahlers 2011a,b).Although is very important in nonrotating convection (Shishkina 2021, Ahlers et al. 2022), the argument has been made (e.g., Liu & Ecke 1997, Julien et al. 2012a, Kunnen 2021) that from the nonhydrostatic quasi-geostrophic perspective, should not play a large role provided the lateral horizontal scale /H is much less than D/H = , which is well satisfied in most RRBC experiments given ∼ Ek 1/3 .Thus, tall thin experiments have been constructed with 1/20 ≤ ≤ 1/2 (Cheng et al. 2018) (Figure 3).This assumption has been challenged by the unexpected robustness of wall mode/BZF states that contribute more strongly for small (de Wit et al. 2020;Zhang et al. 2020Zhang et al. , 2021;;Ecke et al. 2022), approximately as ∼ −1 , and that centrifugal effects scale with H (Horn & Aurnou 2018).An example of the BZF contribution is that the increasing slope of Nu with increasing from Lu et al. ( 2021) is well explained by a decreasing wall mode/BZF contribution ∝ −1 .An important note here is that for any finite , even for â 1, numerical solutions for periodic BCs at the lateral boundaries of the computational domain are different from the solutions for experimental BCs.Although the difference vanishes in the limit → ∞, in realistic simulations with periodic BCs, is relatively small, which causes significant differences in results compared to experiments or DNS with experimental BCs.

Hypothesis-Testing: Comparison of Measurements and Simulations
Let us now consider examples of heat transport data for RRBC in cylindrical containers with ≈ 1/2 using working gases He, N 2 , or SF 6 for 0.7 ≤ Pr ≤ 0.9 and water for 4 ≤ Pr ≤ 6.In Figure 7, the data are plotted in two classical ways: at constant Ra and varying Ek (or Ro) (Figure 7a,c) and at a constant Ek and varying Ra (Figure 7b,d).In most models Nu ∼ Ra γ is assumed with γ = 1/3 in the buoyancy-dominated regime, whereas the proposed values of ξ in Nu ∼ (Ra/Ra c ) ξ in the rotation-dominated regime are different.For example, assuming that heat flux q is independent of H (or of ν and κ), one obtains ξ = 3 (or ξ = 3/2).

Scaling Theory for Heat Transport in Turbulent RRBC
Here we develop a unifying scaling approach for the transition from rotation-dominated to buoyancy-dominated regimes in turbulent RRBC.Summarizing different approaches to estimate heat transport scalings in these regimes, we identify two main ideas.First, in the rotationdominated regime we have Nu − 1 ∼ (Ra/Ra c ) ξ , and in the buoyancy-dominated regime we have Nu − 1 ∼ Ra γ for ξ > 0 and γ > 0. Second, there is a balance of the thicknesses of the viscous Ekman BL, δ E /H ∼ Ek 1/2 , which is thinner in the rotation-dominated regime, and of the thermal BL, δ θ /H ∼ Nu −1 , which is thinner in the buoyancy-dominated regime.Usually considered as independent, these ideas are, however, naturally connected because the change of the scaling regimes in RRBC is related to the change in the BL thickness balance.We thus combine these two ideas, leaving the exponents ξ and γ unspecified, and assume that in turbulent regimes Nu is large (i.e., Nu 1).These arguments give us the following relations, valid for the transition from rotation-dominated to buoyancy-dominated regimes: 10.

Nu
There are some caveats to our scaling approach.First, the data for the wall mode regime (seen in the very left of Figure 10b) are ignored in the present analysis.Here we assumed Nu = 1 at the onset of bulk convection, whereas wall mode convection, which occurs prior to the bulk convection, can lead to Nu values significantly larger than 1 at Ra c in the case of small-containers (Ecke et al. 2022, Zhang et al. 2021).Nevertheless, a reduction by 1 or slightly more in the Nu values in plots like Figure 10a would not significantly influence the scalings, provided Nu 1.Second, the values of γ that are chosen to represent the Nu data for small and large Pr in Figure 10b and Figure 10c, respectively, are, of course, not universal.Here they are chosen empirically, as they better represent the data from Figure 7.The values of γ in RRBC vary, in general, between 3/11 and 3/8 and are larger for larger Ra.To predict the value of ξ in a particular case, one should first estimate γ in the nonrotating/buoyancy-dominated regime [using the theory of Grossmann & Lohse (2000, 2001)] and then calculate ξ using Equation 11.The data should then collapse onto one master curve if plotted as suggested in Figure 10a.Third, the maximal exponent γ = 3/8 in buoyancy-dominated RRBC does not contradict the maximal γ → 1/2 in nonrotating highly turbulent RBC.In the latter case [as in the regime IV l of Grossmann & Lohse (2000)], the velocity BL becomes thinner than the thermal BL, which determines the buoyancy-dominated regime in RRBC.Finally, and Fr should also affect the scaling relations, and this needs further investigation.For the data discussed here, Fr is less than 0.15 for both considered Pr ranges, 0.7 ≤ Pr ≤ 0.9 and 4 ≤ Pr ≤ 6.Additionally, is an influential parameter that can shift the principal scaling regimes within the parameter plane (Shishkina 2021, Ahlers et al. 2022), although it acts in a different manner for RRBC owing to wall mode/BZF contributions and to the decreasing horizontal length scale with increasing rotation.Here we mainly discuss the case of = 1/2.Measurements and DNS data of RRBC in water for a broad range of are given by Lu et al. (2021) and Hartmann et al. (2022).

Non-Oberbeck-Boussinesq Effects
The validity of the OB approximation in RBC was studied by, for example, Spiegel & Veronis (1960) and Veronis (1962), but most comprehensively by Gray & Giorgini (1976).To derive Equations 1-3 from the continuity, momentum, and energy equations for a Newtonian fluid with zero second viscosity (Batchelor 1967), one assumes (a) that all fluid properties are constant except the density in the buoyancy force term in the momentum equation, which is taken to be linearly dependent on the temperature, and (b) that the pressure work and the viscous dissipation terms in the heat equation are negligible.The OB validity means that all terms in the residual equations are negligible.Taking a certain small threshold for the residuals, from assumptions a and b one derives the region of the OB validity in terms of the upper bounds for and H, respectively.The OB validity region for any common fluid is sketched in Figure 4a.Thus, within the OB validity region for any reasonable threshold, both and H < C are bounded, where C depends on the fluid properties alone.This means that for any chosen fluid, Ra larger than a certain value will no longer satisfy the OB criterion.This is a problem that RRBC and nonrotating RBC share.
Non-OB effects in RRBC in water, where the fluid properties are considered temperature dependent, have been studied by Horn & Shishkina (2014).Without rotation, the non-OB effects lead to a global asymmetry of the flow, which is reflected, in particular, in an increased bulk temperature.With increasing rotation, the central temperature approaches that of the OB case, but the asymmetry of the BLs remains.

Centrifugal Buoyancy Effects
Centrifugal buoyancy changes the flow structure and the response characteristics in RRBC (Homsy & Hudson 1971).This holds for both the buoyancy-dominated and rotation-dominated regimes.For example, in a weakly nonlinear rotation-dominated regime, the complex Ginzburg-Landau equation (van Saarloos & Hohenberg 1992, Aranson & Kramer 2002) predicts the scalings of the correlation length as ∼ϵ −1/2 and precession frequency as ∼ϵ 1 , which is also consistent with the numerical solutions of the Swift-Hohenberg equation (Cross et al. 1994).Measurements by Hu et al. (1995Hu et al. ( , 1998) ) for large = 46 and 80, however, deviate from these theoretical predictions, suggesting scaling exponents that are about two times smaller in both cases.Simulations by Becker et al. (2006) have clarified this discrepancy: They show that if the centrifugal term is removed from the momentum equation, the numerical results are consistent with theory (which neglects the centrifugal buoyancy), but inclusion of the centrifugal term leads to results consistent with experiments.Note that the derivation of an OB amplitude equation that includes the centrifugation is nontrivial, as the required toroidal-poloidal decomposition cannot adjust the radial dependency (see Küppers & Lortz 1969, Knobloch 1998, Marques et al. 2007, Scheel 2007, Scheel et al. 2010).
In DNS by Horn & Aurnou (2018, 2019, 2021) and in experiments by Hu et al. (2021Hu et al. ( , 2022)), centrifugal buoyancy effects were investigated over broad ranges of Ro and Fr.Centrifugal buoyancy causes warm (cold) fluid near the bottom (top) plate to move inward (outward) from the centerline with downward flow at the sidewalls, which leads to strongly asymmetric mean temperature profiles in the vertical and radial directions: In the core part of the domain the fluid is always warmer along the centerline than it is near the sidewalls (Hart 2000, Horn & Aurnou 2019).These effects increase with increasing Fr.Horn & Aurnou (2018, 2019) suggested different rotation-dominated regimes in centrifugal buoyancy where the flow can be quasi-geostrophic or quasi-cyclostrophic such that the primary force balance is between the pressure gradient and the Coriolis force or centrifugal buoyancy, respectively.In the cyclostrophic state, tornado-like large-scale structures can form.A triple balance between pressure gradient, Coriolis, and centrifugal forces gives the so-called gradient wind balance, which is particularly important in tropical cyclones (Willoughby 1990).
Different regimes of the dominance of gravitational or centrifugal buoyancy, or of Coriolis forces, can be extracted by analyzing the corresponding timescales τ ff , τ c , and τ in each regime (see the sidebar titled Dimensional Characteristics of Rotating RRBC): The smallest timescale determines the dominance of the corresponding force.Transitions between the regimes are determined by equating the timescales of the neighboring regimes.This way one obtains regime diagrams, as in Figure 11.The dominance of centrifugation over buoyancy is expected for  Fr > /2 (the transition is marked in Figure 11).For a given cell height H, only the rotation rate determines the onset of centrifugal dominance, and not the aspect ratio , since Fr > /2 implies 2 H/g > 1. Experiments by Hu et al. (2022) show, however, that the centrifugal effects become apparent much earlier (at Fr ∼ Ra 0.5 ); this is indicated in Figure 11b.The dominance of the centrifugal over Coriolis force occurs for Ro c > 1 (Figure 11), which means that this region lies outside the region of the OB validity, given Ro c = √ α /2.Thus, to study in DNS regimes dominated by centrifugation, one is forced to consider non-OB governing equations.Another feature is that the gravitational buoyancy time considered here is universally based on H and the free-fall velocity u ff .This implies that for any small Fr, the transition from the rotation-dominated to the buoyancy-dominated regime always takes place at a constant Ro −1 .As we have seen in Section 3.3, however, this is not universal; thus a further inspection of the phenomena of centrifugal effects in RRBC is needed.convective zone of stars.We are partway along that path, but incorporating more realistic governing equations, geometries, and boundary conditions and exploring a broader parameter range remain exciting challenges for the future.
2. The experimental challenges for the future remain being able to fully probe the domain of quasi-geostrophic RRBC while maintaining the asymptotic conditions associated with the model.Better experiments are being developed, and the measurement tools are becoming increasingly powerful.DNS continue to push the envelope of extended range and flexibility.The next decade should be an exciting one for advances in RRBC.

Indexes
Figure 1 (a) An RRBC setup: A container of height H is filled with a fluid and rotated about a vertical axis at angular rate .The bottom (top) is kept at temperature T + (T − ) with T − < T + .(b) Nu versus Ra for experimental runs at constant Ek and Pr ≈ 7. The right arrow indicates decreasing Ek.The Nusselt number in the nonrotating case, Nu 0 ∼ Ra 0.3 , is labeled with small differences in Nu 0 for each data set.(c) Nu/Nu 0 versus Ro for runs at constant Ra ≈ 10 8 .(d-i) Experimental images (Cheng et al. 2020) showing representative examples of (d-f ) rotation-dominated, (g) rotation-affected, and (h-i) buoyancy-dominated or nonrotating states.Panels d-i adapted from Cheng et al. (2020) with permission; copyright 2020 American Physical Society.
Figure 2 (a) Cellular flow showing fluid parcel motion with anticyclonic and cyclonic vertical vorticity that reverses sign at mid-plane.Panel adapted from Veronis (1959) with permission; copyright 1959 Cambridge University Press.(b) Vorticity production (not to scale).Thermal instability pulls converging warm fluid laterally into an expanding plume where it spins up cyclonically, whereas cool diverging return flow from the top (or interior) produces anticyclonic vorticity.Thermal (δ θ ) and kinetic (δ u ) boundary layer (BL) thicknesses are indicated.(c) Vertical profiles of BL structure.Warm (cool) upward (downward) plumes generate positive (negative) cyclonic vorticity from inflowing fluid.The vorticity produced is fairly independent of z outside of the Ekman BL and produces Ekman pumping with the vertical velocity u z ∼ EkHω z at the distance from the plate z = δ u ∼ Ek 1/2 H, where ω z is the vertical component of the vorticity and H is the cell height.Vorticity is dissipated in the Ekman BL per linear Ekman BL processes.The thermal (kinetic) BL thickness is defined by where root-mean-square (rms) temperature T rms (velocity u rms ) is maximum with respect to z.The thermal wind layer (TWL) is in quasi-geostrophic balance with significant Ekman pumping and differs from the thermal BL in nonrotating RBC.
Figure 4 (a) Oberbeck-Boussinesq (OB) validity region in RBC in terms of the maximal temperature difference and container height H, with the cell aspect ratio.(b) Accessible range in a rotating RBC experiment with fixed H in terms of and rotation rate .The text near the restricting lines explains the origins of the restrictions.

Figure 7
Figure 7Typical ways to present heat flux measurements in RRBC: (a,c) for constant temperature difference between the plates (or Ra) and varying rotation rate (or Ek −1 , Ro −1 ) and (b,d) for constant (or Ek −1 ) and varying (or Ra).All shown data are for cylindrical containers with aspect ratio ≈ 1/2 and (a,b) gases He, N 2 , or SF 6 (0.7 ≤ Pr ≤ 0.9) or (c,d) water (4 ≤ Pr ≤ 6).Abbreviation: DNS, direct numerical simulations.Thus, a larger exponent γ in the buoyancy-dominated regime leads to a larger exponent ξ in the rotation-dominated regime.This relation is illustrated in, for example, Figure7c: Larger values of Nu in the buoyancy-dominated region at small Ro −1 indicate more efficient heat transport with larger values of γ , leading to a steeper decrease of the Nu values in the rotation-dominated regime at large Ro −1 .
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