Connections Between Non-Rotating, Slowly Rotating, and Rapidly Rotating Turbulent Convection Transport Scalings

Connections between Non-Rotating, Slowly Rotating, and Rapidly Rotating Turbulent Convection Transport Scalings

Jonathan M. Aurnou∗ University of California at Los Angeles, Los Angeles, California 90095-1567 USA

Susanne Horn Coventry University, Coventry CV1 5FB, United Kingdom

Keith Julien University of Colorado at Boulder, Boulder, Colorado 80309, USA (Dated: September 9, 2020) In this study, we investigate and develop scaling laws as a function of external non-dimensional control parameters for heat and momentum transport for non-rotating, slowly rotating and rapidly rotating turbulent convection systems, with the end goal of forging connections and bridging the various gaps between these regimes. Two perspectives are considered, one where turbulent convection is viewed from the standpoint of an applied temperature drop across the domain and the other with a viewpoint in terms of an applied heat flux. While a straightforward transformation exist between the two perspectives indicating equivalence, it is found the former provides a clear set of connections that bridge between the three regimes. Our generic convection scalings, based upon an Inertial-Archimedean balance, produce the classic diffusion-free scalings for the non-rotating limit (NRL) and the slowly rotating limit (SRL). This is characterized by a free-falling fluid parcel on the global scale possessing a thermal anomaly on par with the temperature drop across the domain. In the rapidly rotating limit (RRL), the generic convection scalings are based on a Coriolis-Inertial- Archimedean (CIA) balance, along with a local fluctuating-mean advective temperature balance. This produces a scenario in which anistropic fluid parcels attain a thermal wind velocity and where the thermal anomalies are greatly attenuated compared to the total temperature drop. We find that turbulent scalings may be deduced simply by consideration of the generic non- dimensional transport parameters — local Reynolds Re` = U`/ν; local Péclet P e` = U`/κ; and Nusselt number Nu = Uϑ/(κ∆T/H) — through the selection of physically relevant estimates for length `, velocity U and temperature scales ϑ in each regime. Emergent from the scaling analyses is a unified continuum based on a single external control parameter, the convective Rossby p 2 number RoC = gα∆T/4Ω H, that strikingly appears in each regime by consideration of the local, convection-scale Rossby number Ro` = U/(2Ω`). Thus we show that RoC scales with the local Rossby number Ro` in both the slowly rotating and the rapidly rotating regimes, explaining the ubiquity of RoC in rotating convection studies. We show in non-, slowly, and rapidly rotating systems that the convective heat transport, parametrized via P e`, scales with the total heat transport parameterized via the Nusselt number Nu. Within the rapidly-rotating limit, momentum transport arguments generate a scaling for the system-scale Rossby number, RoH , that, recast in terms of the total heat flux through the system, is shown to be synonymous with the classical flux-based 2 ‘CIA’ scaling, RoCIA. These, in turn, are then shown to asymptote to RoH ∼ RoCIA ∼ RoC, demonstrating that these momentum transport scalings are identical in the limit of rapidly rotating turbulent heat transfer.

Popular Summary rotating and rapidly rotating buoyancy-driven convective environments. We find that slowly and rapidly rotating Buoyancy-driven convection is likely the dominant scalings can be inter-related via one parameter, the so- driver of turbulent motions in the universe, and thus, called convective Rossby number RoC, a dissipation-free is widely studied by physicists, engineers, geophysicists parameter measuring the importance of buoyancy driving arXiv:2009.03447v1 [physics.flu-dyn] 7 Sep 2020 and astrophysicists. Maybe unsurprisingly, these differ- relative to rotation. Further, we map between non-flux- ent communities discuss the gross convective behaviors based and the flux-based, buoyancy-driven scalings used in different ways, often without significant cross-talk ex- by different groups. In doing so, these scalings show that isting between them. Here, we seek to draw connections there are clean connections between the different com- between these communities. We do so by carrying out a munities’ approaches and that a number of the seemingly set of basic scale estimations for how heat and fluid mo- different scalings are actually synonymous with one an- mentum transport should behave in non-rotating, slowly other.

∗ [email protected] 2

I. INTRODUCTION [e.g., 11, 96]. Other effects are not considered here, such as those due to magnetic fields [71], centrifugal buoy- Accurate parameterizations are ubiquituously sought ancy [55], and non-Oberbeck-Boussinesq-ness [57]. In the for the turbulent transport properties of fluid dynami- Navier-Stokes equation (1a), the velocity vector is u; the cal systems. In buoyancy-driven convection systems, the angular rotation velocity along the axial coordinate z is heat and momentum transport properties are the main Ω; p is the modified pressure; α is the thermal expansiv- focii of such investigations [1, 44, 73, 84]. These trans- ity; g is the gravity vector; and ν is the fluid’s kinematic port estimates are essential for understanding the pos- viscocity. sible behaviors of a given system, and for extrapolating Temperature is T = T + θ where the overbar denotes these behaviors to extreme industrial, geophysical and as- averaging over surfaces perpendicular to g. Thus, T is trophysical settings that are difficult to simulate directly the ‘laterally averaged’ temperature and θ is the temper- [e.g., 14, 27, 32, 41, 50, 54, 88, 99, 105]. ature fluctuation. Equation (1b) is the fluctuating temperature evolution equation and (1c) describes the evo- In the Rayleigh-Bénardconvection systems considered lution of the laterally averaged temperature field. In the here, warmer fluid is maintained at the base of the fluid fluctuating temperature equation (1b), we use the abbre- layer and colder fluid is maintained at the top of the 0 layer, defined with respect to the gravity vector g that viated notation (u · ∇θ) = u · ∇θ − ∇ · (uθ). Convective is parallel to the background temperature gradient. In motions in this system are driven by an unstable, system addition, our system is rotating at angular velocity Ω scale temperature gradient ∂gT = O(∆T/H) measured in the direction of gravity eˆg, where ∆T is the tempera- that is oriented in the axial eˆz-direction. This system is gravitationally unstable, and drives buoyant convective ture drop across the fluid layer of system depth H. Here flows across the fluid layer that advect both heat and ∆T is sustained either via fixed temperature boundaries momentum. We describe this system generally through- or via an applied heat flux Q [cf. 13]. Depending on the out this paper, but it can be thought of as an extended set-up, eˆg can be oriented in the axial direction eˆz [e.g., plane layer [64], a finite cylinder [18], or a spherical shell 20], the cylindrically radial direction eˆs [e.g., 39], or the of fluid [80]. spherically radial direction eˆr [e.g., 80]. Here we take the characteristic convective velocity to A scaling analysis is presented using generic scales for be U, the characteristic length scale to be `, and the char- the characteristic fluid properties occurring in the non- acteristic temperature anomaly to be ϑ. The slowly ro- rotating, slowly rotating and rapidly rotating turbulent tating limit (SRL) is defined such that the inertial forces limits. This analysis generates a large-scale, free-fall flow greatly exceed the Coriolis force: regime in the non-rotating and slowly rotating limits, and a small-scale, thermal wind flow in the rapidly rotating U 2 u · ∇u 2Ω × u −→ 2 ΩU. (2) limit. The generic nature of our scaling analysis allows ` us to provide connections between the different regimes. For instance, we show that the convective Rossby num- The ratio of these terms, the so-called local Rossby number, RoC, arises ubiquitously in scaling estimates for tur- ber defined with the characteristic scales of the convec- bulent rotating convection, both in the rapidly rotating tion, is and slowly rotating end-member limits. Further, Ro C U is shown to be equivalent to Ro`, which describes the Ro` ≡ 1 . (3) Rossby number for the rotating flow dynamics on the 2Ω` local convective scale. The rotating scalings developed In the rapidly rotating limit (RRL) of Rayleigh-Bénard show how numerous heat and momemtum transport laws convection, the Coriolis forces dominate over the inertial can all be inter-related via integer powers of RoC (or, syn- forces, onymously, Ro`), thus providing novel ties between the different scaling regimes. U 2 u · ∇u 2Ω × u −→ 2 ΩU (4) ` Thus, II. GOVERNING EQUATIONS AND U PARAMETERS Ro ≡ 1 . (5) ` 2Ω` The governing equations of rotating thermal convec- We note then that the local Rossby number estimates the tion in a Boussinesq fluid are strength of inertial advection using the estimated convective velocity and length scales considered, normalized by 2 ∂tu + u · ∇u + 2Ω × u = −∇p + gαθ + ν∇ u, (1a) the Coriolis acceleration. 0 2 We are interested in ascertaining turbulent scaling laws ∂tθ + (u · ∇θ) = (eˆg · u)∂gT + κ∇ θ, (1b) for the heat transported across the system scale H and for 2 ∂tT + ∇ · (uθ) = κ∇ T, (1c) the local momentum and heat transport carried by the ∇ · u = 0 (1d) fluid motions occurring on the convective scale `. The 3 system-scale heat transport is measured by the Nusselt where Racrit is the critical Rayleigh number above which number buoyancy-driven fluid motions first onset in a given convection system [15, 65, 106]. Thermal convection is QH UϑH Nu = ∼ , (6) active whenever Raf ≥ 1. No convection occurs for ρcP κ∆T κ∆T Raf < 1, unless a subcritical branch also exists giving where ρ is the fluid’s density and cP its specific heat rise to a hysteretic bistable state. This has have been capacity. Here Q ∼ ρcP Uϑ is the total (superadiabatic) found in low P r, rapidly rotating convection studies in heat flux per unit area, which we assume is dominated spheres, such as in Guervilly and Cardin [48], Kaplan by the turbulent convective transport component (i.e., et al. [69]. The critical Rayleigh number is approximately Nu 1). The momentum and heat transported on the 103 in non-rotating systems [e.g., 93]. More specifically, characteristic convective scale is estimated via the local Racrit = 1708 for no-slip mechanical boundary condi- Reynolds and Pécletnumbers tions in a non-rotating, horizontally-infinite layer of fluid. In contrast, in a rotating plane layer of P r & 0.67 fluid, U` U` the critical Rayleigh number is a strong function of the Re` = , P e` = . (7) ν κ rotation rate and fluid viscosity: The Nu, Re and P e transport scalings will be formu- ` ` Ra ' 8.7 E−4/3 , (14) lated in terms of equations (1)’s non-dimensional control crit parameters, which are the Prandtl, Rayleigh and Ekman and convection onsets in the form of stationary modes. In numbers [e.g., 84]. The Prandtl number describes the lower Prandtl number fluids such that P r . 0.67, con- fluid’s thermophysical properties, vection first develops via oscillatory modes [5, 15, 56] ν and the critical Rayleigh number in a plane layer is P r = , (8) −4/3 κ Racrit ' 17.4 (E/P r) [16, 63]. Thus, in plane layer geometries, Racrit depends on the rotation rate and the where κ and ν are the thermal diffusivity and kinematic fluid’s thermal diffusivity in low P r fluids. In rotating viscosity, respectively. The Ekman number, E gives spherical geometries, the onset is always to P r-dependent the estimated ratio of system-scale viscous and Coriolis oscillatory convection [60]. Although P r can affect Racrit forces: in rotating fluids [e.g., 25], it does not affect the out- ν E = . (9) come of our analyses since all the diffusion coefficients 2ΩH2 drop out of the final expressions. For simplicity, then, we will choose to consider only the moderate P r rela- The Rayleigh number estimates the strength of the buoy- −4/3 ancy forcing tionship Racrit ∼ E from here onwards. Lastly, we present the convective Rossby number, RoC, gα∆TH3 which arises ubiquitously in studies of rotating convec- Ra = . (10) νκ tion. This non-dimensional parameter estimates the ratio of buoyancy and Coriolis forces and is commonly defined The non-dimensional bouyancy forcing will also be pre- as sented in three alternative forms. The first of these is r r in terms of the flux Rayleigh number, based on the heat gα∆T RaE2 Ro ≡ = . (15) flux through the system, C 4Ω2H P r gαQH4 The convective Rossby number is taken to be the essen- RaF = Ra Nu = 2 . (11) ρcP νκ tial control parameter in many studies of rotating convection [e.g., 40, 54, 67, 74, 98, 104, 107], and is also Following Christensen [22] and Christensen and Aubert claimed to control numerous transitions in rotating con- [24], the second form is given in terms of the rotating, vection behavior [e.g., 6, 35, 37, 58, 70, 78, 92]. Further, diffusivity-free, so-called ‘modified flux Rayleigh num- in many rotating convection and dynamo studies, the ber,’ buoyancy forcing is parameterized in terms of the square 3 of the convective Rossby number, although it is referred ∗ RaF E gαQ ∗ 2 RaF = 2 = 3 2 . (12) to there as the ‘modified Rayleigh number’, Ra = RoC P r 8ρcP Ω H [e.g, 22]. (Although this convention is not used here, the oceanic and atmospheric communities write these flux-based pa- Parameter Surveys rameters in terms of the buoyancy flux B = gαQ/(ρcP ) and the Coriolis parameter f = 2Ω [e.g., 81]. Then 4 2 ∗ 3 2 Within the fluid physics community, rotating convec- RaF = BH /νκ and RaF = B/f H .) The third form is the convective supercriticality, tion studies often take the non-rotating limit (NRL) as their philosophical starting point. This assumes an in- Raf = Ra/Racrit , (13) ertial velocity scale and then the inertial turbulence is 4

FIG. 1. Rotating convection heat transport surveys carried out in the slow rotating and rapidly rotating frameworks. a) Slowly rotating approach: Fixed Ra data shingles from the numerical study of Horn and Shishkina [58]. For each data shingle, the vertical temperature difference ∆T (∝ Ra) is fixed and the angular rotation rate Ω (∝ E−1) is varied. b) Rapidly rotating approach: Fixed E data shingles from the laboratory-numerical study of Cheng et al. [19]. Here Ω is fixed and ∆T is varied along each data shingle. The colored x-symbols mark each E-shingle’s Racrit value. The corresponding non-rotating heat transfer efficiency is denoted by Nu0.

perturbed with increasing rotational effects. Within this uses Raf = Ra/Racrit = 1 as its philosophical starting buoyancy-dominated framework, surveys are carried out point, and then perturbs the system with ever increasing at various fixed values of the buoyancy forcing, e.g., fixed values of Raf . In these studies, RoC is not used as a con- Ra ∝ ∆T , while the angular rotation rate of the system trol parameter, but is often checked a posterio to see if it Ω is systematically increased [e.g., 104, 107, 108]. An can collapse the data [e.g., 19, 20, 31, 35, 52]. example of this approach is shown in Figure 1a, which is The two panels of Figure 1 are qualitative mirror im- adapted from the numerical investigation of Horn and ages of one another. Starting from different ends of Shishkina [58]. Six different cuts through parameter the inertially- versus rotationally-dominated ranges, they space are shown, with each data “shingle” made at a show nearly identical data but harvested along different fixed Ra value as shown in the legend box [e.g., 17]. The slices through the same parameter spaces. Figure 1a as- control parameter displayed along the abscissa is 1/RoC, sumes a high Ra, slowly rotating limit (SRL) dominated which in this case varies only as a function of the non- by buoyancy effects, whereas Figure 1b assumes a low −1 dimensional rotation rate of the system E ∝ Ω. The E, rapidly rotating limit (RRL) dominated by Coriolis ordinate shows the Nusselt number, Nu, normalized by forces. its NRL value at each Ra-value, Nu0(Ra). The goal of this study is to develop transport scalings that we bridge the gaps between the NRL, SRL and RRL In the geophysical and astrophysical fluid dynamics convective world views. A particularly important finding communities, it is typically argued that convection occurs is the relative importance of the free-fall terminal velocity within the rapidly rotating limit (RRL) [e.g., 10]. With in the non-rotating and slowly rotating limits and of the this guiding principle in mind, the Ekman number is typ- thermal wind terminal velocity in the rapidly rotating ically fixed at some low value whilst Ra is varied along limit, and how these velocities are related to one another each data shingle. Figure 1b, which is adapted from the via Ro . laboratory-based study of Cheng et al. [19], shows this C approach well. Three different fixed Ekman number shingles are shown. Rayleigh number values are shown on the x-axis and the y-axis denotes the Nusselt number values. III. THE NON-ROTATING AND SLOW ROTATING LIMITS (The solid black line denotes the NRL scaling Nu0(Ra).) −4/3 Small x’s on the abscissa denote RaCrit = 8.7E , the critical Ra value at which stationary planar rapidly rotat- In the limits of asymptotically high Ra, high Re, tur- ing convection onsets at a given E value. Such a survey bulent convection, we presume that perfect power law 5 scaling behaviors exist to describe the heat and momen- arrive at the standard, time-scale based description of the tum transport in terms of the other relevant system pa- convective Rossby number, RoC = τΩ/τff , as the ratio of rameters, Nu(Ra, P r) and Re(Ra, P r) [cf. 21, 45, 98, the rotational time, τΩ = 1/(2Ω), and the free-fall time 100]. The demonstration of such asymptotic scalings is across the system-scale. still an active and frothy topic of scientific debate [e.g., The scales in (17) lead to the following NRL and SRL 29, 42, 53, 75, 77, 109]. We assume, further, that similar transport estimates transport scalings exist in the non-rotating and slowly U` U H pgα∆TH3 rotating regimes. Despite, small differences due to sym- Re = ∼ ff = metry breaking in slowly rotating systems [e.g., 9, 107], ` ν ν ν their gross transport behaviors can be taken to be com- Ra1/2 parable (e.g., Figure 2). = ≡ Reff , (20a) P r For both non-rotating and slowly rotating convection, we take the characteristic convection length scale to be U` U H pgα∆TH3 P e = ∼ ff = the global scale of the system in all directions, ` ∼ H, ` κ κ κ based on the superstructures that form at high Ra with 1/2 = (Ra P r) ≡ P e , (20b) vertical scales of order H and lateral scales that are typ- ff ically less than 10H [e.g., 2, 83, 90, 102, 103], which ap- pear to be maintained even in extreme astrophysical and UϑH U H Nu ∼ ∼ ff = (Ra P r)1/2 ≡ P e . (20c) geophysical systems [e.g., 87]. In the turbulent limit, the κ∆T κ ff free-fall inertial balance is achieved U 2 Here, Re and P e are the classic diffusivity-free, free- u · ∇u ∼ gαθ −→ ∼ gαϑ. (16) ff ff H fall transport parameters. Analytic estimations for the characteristic magnitude of Dimensional analysis can be used, independently, to ϑ in the turbulent regime are non-trivial [e.g., 1, 100]. solve for the coefficients ζ and χ that yield diffusivity-free Here, following the work of Grossmann and Lohse [e.g., expressions for the characteristic transport parameters 47], we scale ϑ ∼ ∆T . In the NRL and SRL, it then [e.g., 84], yielding follows that ζ χ 1/2 ReH ∼ Raf P r = (Ra/P r) ≡ Reff p ` ∼ H, ϑ ∼ ∆T and U ∼ gα∆TH ≡ Uff . (17) (ζ = −χ = 1/2), (21a) ζ χ 1/2 The dominant flows in these regimes are large scale; these P eH ∼ Raf P r = (Ra P r) ≡ P eff flows are driven by thermal fluctuations that are roughly (ζ = χ = 1/2), (21b) comparable to the temperature drop across the system ζ χ 1/2 (likely akin to the characteristic boundary layer tem- Nu ∼ Raf P r = (Ra P r) ≡ P eff perature variations); and the convective flows will ap- (ζ = χ = 1/2), (21c) proach Uff , the diffusivity-free, inertial free-fall velocity [e.g., 21, 94, 95]. Further, the characteristic advective where Raf 7→ Ra in the dimensional analysis, since Racrit time scales are isotropic and are given by is effectively constant in the non-rotating and slowly rotating limits. Since it is being assumed that the convec- ` H p tion is highly supercritical and turbulence dominated, we τU = ∼ = H/(gα∆T ) ≡ τff , (18) U Uff take (Nu − 1) ≈ Nu,(Re − 1) ≈ Re, and (Raf − 1) ≈ Raf in all our dimensional analyses [cf. 30]. where τff is the inertial free-fall time across the system. The dimensional analytical transport estimates in (21) We note, following Spiegel [94], that our assumption that are consistent with the dynamical scaling estimates given transport processes are dominated by the large scale flows in (20), and also agree with the classic dimensional analy- likely best applies in low P r fluids [e.g., 102]. We will sis predictions for non-rotating convection in the limit of not probe this assumption more deeply here, but direct zero diffusive effects [95]. The agreement between the in- readers to more focussed treatments of non-rotating RBC dependent scalings (20) and (21) shows that Re ∼ Re [e.g., 1, 28, 46, 100]. ` H and P e` ∼ P eH , consistent with our assumption that Using the SRL scales (17), the local Rossby number ` ∼ H in NRL and SRL. Lastly, multiplying by E, the can be recast as momentum transport scalings (20a) and (21a) require r that U U gα∆T Ro = ∼ ff = ≡ Ro , (19) ` 2Ω` 2ΩH 4Ω2H C Ro ∼ Ro ∼ Ro (22) which demonstrates that the local Rossby number is ` H C equivalent to the convective Rossby number, RoC, in the slowly rotating limit (Ro` 1). Further, from (19), we in the slowly rotating regime, consistent with (19). 6

FIG. 2. Non-rotating convective flows, which approximate the slowly rotating limit (SRL). a) Temperature field image adapted 8 4 from the Ra = 10 ; P r = 1; radius ratio χ = 0.6 spherical shell simulation of Gastine et al. [36], corresponding to Reff = 10 . Lighter (darker) shading represents warmer (cooler) fluid. b) Laboratory shadowgraph image courtesy of Jewel Abbate (UCLA) 11 5 showing convection in 1.5 cSt silicone oil for Ra ' 4 × 10 and P r ' 21, corresponding to Reff ' 10 . The cylindrical tank is 40 cm high by 20 cm across, with shape distorted and its left hand side clipped by the shadowgraph technique. The horizontal line near the mid-plane and the dark region at the tank bottom are further lighting artifacts.

IV. THE RAPIDLY ROTATING LIMIT (RRL) in which the first term is inertial advection of vorticity (I), the second is the axial stretching of planetary (or Just as angular momentum is the key dynamical vari- background) vorticity (C), and the third is the buoyancy able in rapidly rotating solid mechanics problems, vor- torque (A). ticity, ω = ∇ × u, is the essential dynamical variable in Rapidly rotating convective motions are strongly rapidly rotating fluid systems in which rotational inertia anisotropic, as shown in Figure 3, with small scales per- dominates the physics [43]. The evolution equation for pendicular to Ω and much longer scales parallel to Ω. fluid vorticity, ∇× (1a), is: Therefore, it is essential in (24) to distinguish between the characteristic convection scale ` measured perpendic- ∂tω + u · ∇ω − ω · ∇u = ular to Ω and the system scale H measured parallel Ω. 2Ω · ∇u + ∇ × (gαθ) + ν∇2ω . (23) Only the stretching of the background vorticity, 2Ω ∂zu, can occur on the system scale. The other two terms, I In the turbulent rapidly rotating limit, a balance is and A, operate on the local convective scale. Although achieved in (23) between the inertial (I), Coriolis (C), the length scales ` and H differ greatly in rapidly rotat- and buoyancy (A, for Archimedean) terms [4, 59]. This ing convection, the kinetic energies measured along these is typically referred to as the CIA balance [61, 73]: different directions remain comparable, even in the supercritical regime [58, 68, 96, 97, 101]. Thus, we assume that the characteristic velocity magnitudes are approxi- u · ∇ω ∼ 2Ω ∂zu ∼ ∇ × (gαθ) (24) mately isotropic |ui| ∼ U in RRL. U 2 2ΩU gαϑ ∼ ∼ `2 H ` The balance between the C and I terms in (24) then 7

FIG. 3. Anisotropic flows in rapidly rotating convection with longer characteristic axial scales than horizontal scales ` (measured perpendicular to the rotation axis). a) Temperature field image from the Ra = 2.5 × 109; E = 10−6; P r = 1; radius ratio −2 2 χ = 0.6 spherical shell simulation of Gastine et al. [34], corresponding to RoC = 5 × 10 and ReTW ' 1.3 × 10 . Lighter (darker) shading represents warmer (cooler) fluid. b) Laboratory shadowgraph image courtesy of Jewel Abbate (UCLA) showing 11 −7 −2 rotating convection in 1.5 cSt silicone oil for Ra ' 5 × 10 ; E ' 6 × 10 ; P r ' 21, corresponding to RoC ' 9 × 10 and 3 ReTW ' 1.3 × 10 . The cylindrical tank is 40 cm high by 20 cm across, with its shape distorted and clipped around the mid-plane by the shadowgraph imaging technique. The horizontal line near the mid-plane and the dark region at the tank bottom are further lighting artifacts.

gives This implies, in the RRL, that

` U τ ϑ ` Ω ∼ ∼ Ro` . (27) ∼ ≡ Ro` = ` , (25) ∆T H H 2Ω` τU

Balancing the C and A terms in (24) yields ` where the lateral advective time scale τU = `/U charac- terizes rapidly rotating convection. Thus, rapidly rotat- gα∆T ϑ H gα∆T ing convection is highly anisotropic with ` H, since U ∼ ∼ ≡ UTW , (28) ` ∼ Ro`H in (25) and Ro` 1 in the definition of 2Ω ∆T ` 2Ω the RRL. Unlike in the NRL and SRL where the bulk fluid tends to be isothermalized by strong turbulence, in where UTW , the thermal wind velocity, is the diffusivity- rapidly rotating convection, an unstable mean temper- free velocity scale in the rapidly rotating convection ature gradient tends to be sustained in the fluid bulk, regime [82]. (This thermal wind scaling is similarly found ∂gT ∼ ∆T/H [e.g., 19, 51, 67, 68]. The fluctuating ther- by balancing the I and A terms in (24).) From (28), we mal energy equation (1b) thus scales as see that the local advection time scale in RRL is the thermal wind time scale:

0 Uϑ U∆T (u · ∇θ) ∼ (eˆg · u)∂gT −→ ∼ . (26) ` ` H τU = `/UTW ≡ τTW . (29) 8

The rapidly rotating local Rossby number then becomes where ReTW and P eTW are the thermal wind Reynolds and thermal wind P´ecletnumbers, respectively. UTW τΩ gα∆T 1 The scaling analysis in (35) is consistent with rapidly Ro` ∼ = = 2 2Ω` τTW (2Ω) H Ro` rotating, diﬀusivity-free dimensional analysis, which r yields RaE2 −→ Ro` ∼ ≡ RoC . (30) ζ χ 3/2 2 P r Re` ∼ Raf P r = (Ra/P r) E ≡ ReTW (ζ = −χ = 3/2) , (36a) Thus, the a posteriori local Rossby number, Ro`, is equivalent to the a priori convective Rossby number, RoC, in 3/2 ζ χ Ra 2 both the slowly rotating limit (19) and in the rapidly ro- P e` ∼ Raf P r = E ≡ P eTW P r1/2 tating limit (30). At closer inspection, this holds because (ζ = −3χ = 3/2) , (36b) the local advective time scales, τff = H/Uff in SRL and

τTW = `/UTW in RRL, are similar. Thus, their ratio 3/2 ζ χ Ra 2 yields Nu ∼ Raf P r = E ≡ P eTW P r1/2 τ H U 1 gα∆T/(2Ω) (ζ = −3χ = 3/2) . (36c) ff ∼ TW ∼ √ = O(1) . (31) τTW ` Uff RoC gα∆TH Note that the critical Rayleigh number in (36) varies strongly with the system’s rapid rotation, Raf ∼ RaE4/3. This similarity between the SRL and the RRL local ad- Consistency between (35) and (36) requires that the per- vective time scales explains why the convective Rossby tinent velocity and length scales must be U and ` in number turns up so ubiquitously in rotating convection TW RRL. Thus, Re ∼ Re` ≡ ReTW and P e ∼ P e` ≡ P eTW dynamics: even though Uff and H in SRL both greatly in the rapidly rotating regime. Multiplying (35a) by exceed UTW and ` in RRL, their ratios, Uff /H and 2 the local Ekman number, E` = ν/(2Ω` ), yields Ro` = UTW /` have equivalent scaled values. Expression (31) ReÈ` ∼ RoC, consistent with (30). Further, the RRL demonstrates, further, that the convective Rossby num- heat transport scaling (35c) is also consistent with asym- ber can be cast, alternatively, as potically reduced theory and diffusivity-free formulations [e.g., 7, 66, 79, 84]. Recent studies, such as Plumley et al. UTW RoC ≡ . (32) [85, 86], suggest that it is possible to reach the RRL scal- Uff ings (35) at far lower Raf values than are necessary to This velocity-based definition of RoC holds in both slowly reach their diffusivity-free non-rotating counterparts [cf. rotating and rapidly rotating regimes, and differs in its 28]. interpretation in comparison to the standard (slowly ro- The rapidly rotating thermal wind transport scalings tating) definition in which RoC = τΩ/τff , as will be dis- in (35) differ from the slowly rotating free-fall scalings 2 cussed further in section VI. by a factor of RoC. This creates a clean and novel link In the limit of rapidly rotating convective turbulence, between the two sets of scaling predictions. We can al- the CIA balance gives ternatively cast the RRL expressions as

3 −1 ` ∼ RoCH , ϑ ∼ RoC∆T, (33) Re` ∼ RoC E , and gα∆T P e ∼ Nu ∼ Ro3 (E/P r)−1 . (37) U ∼ Ro U = ≡ U , (34) ` C C ff 2Ω TW From (24), we predict that rapidly rotating turbulent with all three turbulent RRL scales diﬀering by RoC rel- transport data acquired with approximately ﬁxed rota- ative to their corresponding SRL scales. Following the tion rate and material properties will be collapsed when same steps as in (17) but employing the rapidly rotating normalized by the cube of the convective Rossby number. scales in (34) then leads to the following RRL transport Local scale parameterizations naturally arise in our estimates analysis of rapidly rotating transport phenomena. How- 3/2 ever, the system scale transport parameters, ReH and UTW ` gα∆T ` Ra 2 Re` = ∼ = E P e , are most often reported in the literature [e.g, 48]. ν 2Ω ν P r H Thus, we rescale our local rapidly rotating transport scal- 2 = RoCReff ≡ ReTW , (35a) ings to provide the equivalent, system scale counterparts: U ` gα∆T ` Ra3/2 P e = TW ∼ = E2 H U H RaE ` κ 2Ω κ P r1/2 Re = Re ∼ TW = H ` ` ν P r 2 = RoCP eff ≡ P eTW , (35b) −1 = RoC ReTW = RoCReff (38a) 3/2 UTW ϑH UTW ` Ra 2 H U H Nu ∼ = = E P e = P e ∼ TW = RaE κ ∆T κ P r1/2 H ` ` κ 2 −1 = RoCP eff ≡ P eTW , (35c) = RoC P eTW = RoCP eff . (38b) 9

In addition, the system scale Rossby number scales as Rapidly Rotating Flux-Based Scalings Ro = Re E ∼ Ro2 , (39) H H C In order to formulate the flux-based, system-scale, in agreement with the low-E, quasi-geostrophic con- rapidly rotating momentum transport scaling, we recast vection models Guervilly et al. [49] and the three- the RRL heat transport scaling (35c) as dimensional asymptotically-reduced models of Maffei 1/2 −2 2/5 et al. [79]. This system scale RRL Rossby number scal- Ra = (RaNuP r E ) 2/5 ing (39) differs by a factor of RoC relative to the slowly 1/5 −4/5 = RaF P r E . (44) rotating scaling (22) in which RoH ∼ Ro` ∼ RoC. Substituting (44) into (35) leads to the local, flux-based, rapidly rotating transport scalings: V. FLUX-BASED SCALINGS 3/5 Ra E4/3 Re ∼ F TW P r2 Non-Rotating and Slowly Rotating Flux-Based ∗ 3/5 −1 3 −1 Scalings = RaF E = RoCE , (45a)

4/3 3/5 When considering a planetary or stellar convection sys- RaF E P eTW ∼ tem, it is far easier to estimate the outward thermal flux P r1/3 than to infer a temperature drop across a given fluid ∗ 3/5 −1 3 −1 = RaF (E/P r) = RoC(E/P r) , (45b) layer. Therefore, it is of great utility to recast the scal- with the local scale, RRL flux-based Rossby number ings developed above in terms of the (superadiabatic) heat flux, Q, instead of the temperature difference, ∆T . Ra E3 1/5 Ro ∼ F = Ra∗ 1/5 = Ro . (46) Non-dimensionally, this simply corresponds to replacing ` P r2 F C the Rayleigh number, Ra ∝ ∆T , with the flux Rayleigh number, RaF = RaNu ∝ Q. In order to recast the NRL Note, using the flux-based expression for UTW (given in and SRL scalings in terms of RaF , we manipulate (20c) (50) below), that again Ro` ≈ RoC. However, in contrast into the form to the fixed temperature configuration, the flux-based lo- 2/3 cal Rossby numbers in the SRL (42) and the RRL (46) h −1/2i 2/3 −1/3 Ra ∼ RaNuP r ∼ RaF P r , (40) are no longer identical. Instead, both flux-based Ro` expressions depend on the modified flux Rayleigh number, ∗ and substitute this into (20a) and (20b), giving the flux- but in the SRL RaF is raised to the one-third power, based free-fall scalings whereas it is raised to the one-fifth power in the RRL. 1/3 This difference in the flux-based Ro` expressions stems RaF 1/3 from the different Ra(Ra , E, P r) scalings given in (40) Reff ∼ and P eff ∼ [RaF P r] . (41) F P r2 and (44). The SRL flux-based expression for the Rossby number is The system scale, flux-based, rapidly rotating trans- then [cf. 81] port scalings are often used in the geophysical and astrophysical literature [e.g., 23, 39]. These are found by 3 1/3 substituting (44) into (38a), which leads to RaF E ∗ 1/3 Ro` ∼ RoH ∼ 2 = RaF . (42) P r Ra 2/5 Re ∼ F E1/5 ≡ Re , (47a) The respective dimensional forms of the uncontrolled H P r2 CIA temperature drop (which is assumed here to be propor- 2/5 P e ∼ Ra (E P r)1/5 = Re P r . (47b) tional to ϑ in NRL and SRL) and the free-fall velocity H F CIA scale in the slowly rotating regime are The flux-based ReH expression (47a) is referred to as the CIA scaling velocity, Re , since it is indeed derived 2 1/3 CIA Q from the CIA triple balance [e.g., 4, 59, 61, 73]. This flux- ∆T ∼ Q/ρcP Uff ∼ 2 2 , (43a) gαρ cP H based momentum transport scaling is easily converted 1/3 back into a temperature-based scaling by substituting p gαQH U ∼ Uff ∼ gα∆TH ∼ . (43b) RaF = RaNu into ReCIA and then further substituting ρcP 3/2 2 1/2 Nu ∼ Ra E /P r = P eTW . Doing so yields Equations (43) correspond to the free-fall balance ex- 2/5 pressed in terms of an applied heat flux Q [e.g., 26]. Fur- Ra Ra3/2E2 Re ∼ E1/5 = Ro Re , (48) ther, by inserting (43a) into (15), we find that the flux- CIA P r2 P r1/2 C ff based SRL expression for Ro = Ra∗ 1/3, identical to Ro C F ` in agreement with (35a) and (38a). Multiplying (48) by in (42). Thus, Ro` ≈ RoC in the flux-based framework as E then demonstrates that well, as must be the case since this result is framework 2 independent. RoCIA ∼ RoC (when Nu → P eTW ). (49) 10

Thus, the classical, flux-based CIA theory is synony- from making simple, flux-based connections between the mous with the temperature-based rapidly rotating ve- scaling regimes, as is possible in the temperature-based locity scalings given in (38a) and (39). framework. Nevertheless, exploration of the flux-based Since most laboratory and numerical simulations can- framework has shown that the classical, flux-based CIA not reach the diffusivity-free P eTW heat transfer trend, scalings produced in many prior works is formally syn- 2 the RoH ∼ RoC scaling is difficult to attain [cf. 8, 49, 79]. onymous with the temperature-based scalings developed For example, in the seminal planetary dynamo survey of herein (cf. (39) and (49)). Christensen and Aubert [24], it was found that RoH ∼ ∗ 2/5 RaF , which, comparing to (47a), shows that the bulk flow had attained the turbulent, CIA scaling. Their heat VI. DISCUSSION transfer data was best fit as Nu ∼ RaE, which differs from the P eTW scaling likely because it was controlled by The convective scaling relationships presented here diffusive, boundary layer physics [e.g., 17, 52, 62]. This are generated via exactly parallel constructions, first 8/5 corresponds to RoH ∼ RoC . However, if we substitute made within the non-rotating and slowly rotating lim- Nu = Ra3/2E2/P r1/2 in place of their Nu ∼ RaE scal- its and then secondarily made within the rapidly rotating, then the system-scale Rossby number scaling neces- ing limit. Starting from the generic non-dimensional 2 sarily transforms to RoH ∼ RoC. transport parameters, Re = U`/ν, P e = U`/κ and Our flux-based momentum transport scalings show Nu = Uϑ/(κ∆T/H), we select the dynamically relevant that the RRL transport (38a) is formally identical to estimates for `, ϑ and U that characterize a given con- the classical, flux-based CIA velocity scaling (47a) when vection system. Two configurations of thermal driving Nu ≈ P eTW . However, this P eTW heat transfer scal- are considered: the fixed-temperature regime (Table I), ing is not often found in standard experiments or direct popular for its ease of application and interpretation in numerical simulations, because the heat transfer rarely modeling studies, and the fixed heat flux regime (Table reaches the RRL trend [cf. 7, 66, 85, 86]. This is an im- II), popular for its ease of application in geophysical and portant physical point, as the flux-based ReH scaling in astrophysical settings. (47a) can be applied for any Nu value and, accordingly, is The fixed-temperature configuration is particularly el- often considered to be fundamentally different from, and egant, and we will focus on the fixed temperature scalings to conflict with, the local scale prediction (35a) and the in this discussion. First, our analyses show that the lo- system scale prediction (38a) that both naturally arise cal Rossby number is equivalent to the convective Rossby in the Nu ≈ P eTW rapidly rotating scaling turbulent number, arguments given here and in rapidly-rotating asymptotic analysis [10, 12, 79, 96]. Directly comparing the Reynolds Ro` ' RoC, numbers scalings in (35a) and (47a) is, however, incor- rect since they are defined on different length scales. In in both the slowly and the rapidly rotating frameworks, contrast, it is appropriate to compare (38a) and (35a) where Ro` ≡ U/(2Ω`) is estimated using the characteris- since they are both system-scale quantities, and we have tic convective√ length `, the velocity scale U for each limit, 2 −1 shown, in fact, that these scalings are identical in the and RoC ≡ RaE P r . turbulent rapidly rotating limit where Nu → P eTW . Second, by taking the ratios of the rapidly rotating and The respective dimensional forms of the rapidly ro- slowly rotating characteristic scales, we find that they are 1 tating temperature fluctuation, temperature drop, and all related via powers of RoC, velocity scales are: ` ϑ UTW 1/5 3/5 ∼ ∼ ∼ RoC , (51) Q 2Ω Q H ∆T Uff ϑ ∼ ∼ 2 2 , (50a) ρcP UTW g α H ρcp Thirdly, we have shown that the RRL thermal wind ϑH (2Ω)4/5 H1/5 Q 2/5 ∆T ∼ ∼ , (50b) transports and the SRL free-fall transports differ from ` 3/5 ρc 2 (gα) p one another via powers of RoC, p gαϑ H U ∼ UTW ∼ gαϑ` ∼ ReTW P eTW 2 2Ω ` ∼ ∼ RoC . (52) Reff P eff gαQ2/5 H 1/5 ∼ . (50c) ρcp 2Ω Further, our generic scalings predict that the system- In this section, we have transformed the scaling results scale Rossby number, RoH , scales as RoC in the slowly ro- 2 produced in the ∆T -based framework to the Q-based tating regime and as RoC in the rapidly rotating regime. framework through the definition of the flux Rayleigh Thus, the convective Rossby number is shown to explain number RaF = RaNu. However, in the flux-based scal- the local-scale convection dynamics, Ro` ≈ RoC, and is ings, we find a lack of equivalence between the SRL essential for relating the slowly rotating convection be- and RRL local Rossby numbers, thereby preventing us haviors to those of the rapidly rotating regime. Thus, 11

Regime Ro` ` ϑ U Re` P e` Nu ReH P eH RoH (∆T -based) ≈RoC SRL 1 H ∆T Uff Reff P eff P eff Reff P eff RoC 2 2 2 2 RRL 1 RoCH RoC∆T RoCUff RoCReff RoCP eff RoCP eff RoCReff RoCReff RoC

TABLE I. Summary of applied ∆T , turbulent scaling estimates for characteristic convective scales and transports√ in the slowly rotating limit (SRL) and the rapidly rotating limit (RRL). The free fall velocity is deﬁned here as Uff ∼ gα∆TH and p 2 Ro` ∼ RoC = RaE /P r in both SRL and RRL. The non-rotating (NRL) scalings are identical to SRL in our treatment, excepting that the Rossby number is not deﬁned in the non-rotating regime.

Regime Ro` ` ϑ U Re` P e` Nu ReH P eH RoH (Q-based) ≈RoC SRL 1 H ∆T Uff Reff P eff P eff Reff P eff RoC 1/3 4/3 4/3 4/3 1/3 1/3 2 RRL 1 RoCH RoC∆T RoC Uff RoC Reff RoC P eff RoC P eff RoC Reff RoC Reff RoC

TABLE II. Summary of applied Q, turbulent scaling estimates for characteristic convective scales and transports in the slowly 1/3 rotating limit (SRL) and the rapidly rotating limit (RRL). The free fall velocity is defined here as Uff ∼ (gαQH/ρcP ) = 1/3 ∗ 1/3 ∗ 1/5 (BH) . Note in the flux based framework that Ro` ∼ RoC ∼ RaF in the SRL, whereas Ro` ∼ RoC ∼ RaF in the RRL. 2 ∗ 2/5 2/5 For example, we find RoH ∼ RoC ∼ RaF in the rapidly rotating regime, which is consistent with ∆T ∼ Q in (50b).

RoC (and synonymously Ro`) arise ubiquitously in de- Thus, the Rossby number based on the dominant dynam- scribing rotating convective flows. Furthermore, the the- ical scale is equivalent to the convective Rossby number oretical framework we have developed here provides a in both end member rotational regimes, Ro` ' RoC. novel, and remarkably straightforward, set of experimen- This makes clear that the convective Rossby number tally testable interconnections between the slowly rotat- is, in fact, an appropriate descriptor of rapidly rotating and rapidly rotating convective regimes. As summa- ing convection dynamics, but it should always be cast as rized in Table 1, these scalings all depend rather simply RoC = UTW /(2Ω`) in the rapidly rotating limit. Further, on Ro` ≈ RoC. since Ro` ' RoC in both regimes, RoC can be further We have shown that when Ro` is defined using the interpreted as the descriptor of the local scale rotating appropriate slowly rotating characteristic scales is equiv- convection dynamics, irrespective of its value. We con- alent to the convective Rossby number RoC: clude then that the convective Rossby number really ties the room together. r 2 Uff τΩ RaE Ro` = = = ≡ RoC (SRL). (53) 2ΩH τff P r The fixed heat flux configuration can be deduced from the fixed-temperature configuration through the relation Following from this, RoC is often interpreted as the ratio RaF = RaNu. We again find that Ro` ' RoC in both between freely falling convective inertia and the system’s the slow rotating and rapidly rotating limits. However, rotational inertia [e.g., 6, 40, 57, 67]. This interpretation ∗1/3 they no longer have a common definition: RoC ∼ Ra is accurate in the slowly rotating regime [e.g., 9, 35, 107]. F in the SRL regime and Ro ∼ Ra∗1/5 in the RRL regime. In contrast, this U -based interpretation is not accurate C F ff The relationships between the various flux-based scalings in rapidly rotating cases, where the length and velocities are given in Table 2. scales are far smaller than in the slowly rotating regime (Table 1). Irrespective of the configuration, a clear interpretation Surprisingly, though, we have shown that the Ro` also of RoC arises from our scaling analyses. The two charac- scales equivalently to RoC in the rapidly rotating limit: teristic velocities in rotating convection are Uff and UTW . r 2 In slowly rotating convection, U ∼ Uff UTW , since all UTW τΩ RaE Ro` = = = ≡ RoC (RRL). (54) the fluid’s buoyant potential energy is converted to ki- 2Ω` τTW P r netic energy well before it reaches UTW . (Alternatively This equivalence holds since the free-fall time scale in stated, UTW becomes singularly large as Ω becomes the slowly rotating regime scales similarly to the thermal small.) In rapidly rotating convection, U ∼ UTW Uff wind time scale in the rapidly rotating regime, since the vortex stretching term in (23) greatly limits the distance through which a rotating parcel of buoyant fluid H ` can actually freely fall [61]. The selection between Uff τff = ∼ = τTW . (55) Uff UTW and UTW is based on the more restrictive value between 12 the two: devices, and associated state of the art numerical simulations, will allow investigations into the efficacy and U ' min(Uff ,UTW ) . (56) applicability ranges of the turbulent scaling predictions presented here (Tables I and II). Our goal will then be to Since RoC = UTW /Uff , it can be validly interpreted as test, possibly validate, and disambiguate between these the essential control parameter that picks between the differing scaling laws given high fidelity measurements, two characteristic velocitites: and thereby deduce accurate, robust relations for non- Ro 1 ⇒ min(U ,U ) = U , (57a) rotating, slowly rotating, and rapidly rotating convective C ff TW ff heat and momentum transport, as is necessary to explain RoC ∼ 1 ⇒ min(Uff ,UTW ) = U, (57b) and interpret industrial, astrophysical and geophysical RoC 1 ⇒ min(Uff ,UTW ) = UTW . (57c) convection phenomena. The relative ordering of the characteristic time scales is also, therefore, set by RoC: Acknowledgements H RoC 1 ⇒ τΩ (τff ∼ τU ), (58a) ` H RoC ∼ 1 ⇒ τΩ ∼ τff ∼ τU ∼ τU , (58b) This work arose from discussions at the “Rotating Con- ` H vection: From the Lab to the Stars” workshop held at RoC 1 ⇒ τΩ (τff ∼ τ ) τ . (58c) U U the Lorentz Center (https://www.lorentzcenter.nl) The intermediate RoC ∼ 1 regime has not been consid- in May 2018. We gratefully acknowledge the finan- ered here. There is, however, a great deal of laboratory cial support of the NSF Geophysics Program (EAR [e.g., 20, 30, 72, 74, 76, 89, 104] and numerical simula- awards 1620649 and 1853196) and NSF Applied Math- tion data [e.g., 3, 6, 24, 33–35, 38, 40, 67, 78, 91] in the ematics Program (DMS award 2009319), NSF Astro- RoC = O(1) regime. Thus, its scaling behaviors are of nomical Sciences (AST award 1821988), NASA (award broad interest and should be considered in future studies. 80NSS18K1125) and the German Research Foundation An array of new convection and rotating convection de- (DFG award HO 5890/1-1). Further thanks are given to vices have been recently built at research centers world- Jewel Abbate (UCLA) and Thomas Gastine (IPGP) for wide [e.g., 18, 75, 110]. These next-generation laboratory supplying the images used in Figures 2 and 3.

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