Scaling Behavior in Rayleigh-Bénard Convection with and Without Rotation

Scaling behavior in Rayleigh -Bénard convection with and without rotation

Manuscript ID: Draft

mss type: Fast Track

Date Submitted by the Author: n/a

Complete List of Authors: King, Eric; UC Berkeley, Earth and Planetary Science Stellmach, Stephan; Westfälische Wilhelms-Universität Münster, Institut für Geophysik Buffett, Bruce; UC Berkeley, Earth and Planetary Science

Bénard convection < Convection, Rotating flows < Geophysical and Keyword: Geological Flows

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Under consideration for publication in J. Fluid Mech. 1

Scaling behavior in Rayleigh-B´enard convection with and without rotation

E. M. KING1, S. STELLMACH2, A N D B. BUFFETT1

1Department of Earth and Planetary Science, University of California, Berkeley, 94720-4767 USA 2Institut für Geophysik, WWU Münster, AG Geodynamik Corrensstr. 24, Münster, 48149 Germany 3Department of Earth and Space Sciences, University of California, Los Angeles, 90095-1567 USA

(Received soon)

Rotating Rayleigh-Bénard convection provides a simplified dynamical analog for many planetary and stellar fluid systems. Here, we use numerical simulations of rotating Rayleigh- Bénard convection to investigate the scaling behavior of five quantities over a range of Rayleigh (103 . Ra . 109), Prandtl (1 P r 100), and Ekman (10−6 E ). The five quantities of interest are the viscous≤ and≤ thermal boundary layer thicknesses,≤ ≤ ∞ mean temperature gradients, typical horizontal length scales, and flow speeds (Peclet number). We focus, where possible, on scalings with some theoretical basis. Also, the parameter ranges in which different scalings apply are quantified, and so separate dynamical regimes are identified.

Key Words: rotating convection

1. Introduction Convection is a ubiquitous natural process, as fluids throughout the universe flow to redistribute destabilizing thermal and gravitational energy stores. Astro- and geophysical bodies also rotate, and the convective flows in many of these systems are strongly influenced by Coriolis acceleration. We examine a simplified analog of these systems by way of Rayleigh-Bénard convection (RBC) in a Boussinesq fluid with and without rotation. RBC consists of a fluid layer sandwiched between two rigid, horizontal boundaries (Bénard 1900; Rayleigh 1916). The bottom boundary is warmer than the top, and the gravity vector points downward. This temperature drop, ∆T , destabilizes the fluid layer, as cool, heavier fluid lies atop warmer, more buoyant fluid. The convection is treated as Boussinesq in that fluid properties are isotropic and fixed in time and space, except in the buoyancy force, which is driven by thermal expansion. For rotating RBC, the fluid layer is rotated about a vertical axis with constant angular frequency Ω. Rotating RBC in a Boussinesq fluid is governed by the conservation of momentum, Page 2 of 21

2 E. M. King, S. Stellmach, and B. Buﬀett

Parameter Deﬁnition Simulations

Rayleigh Number Ra ≡ (αg∆T L3)/(νκ) 103 . Ra . 109 Prognostic − Ekman Number E ≡ ν/(2ΩL2) 10 6 ≤ E ≤∞ Prandtl Number Pr ≡ ν/κ 1 ≤ Pr ≤ 100

Nusselt Number Nu ≡ (qL)/(k∆T ) 1 . Nu . 50 Output Peclet Number Pe ≡ (UL)/κ 3

Table 1. Relevant non-dimensional parameters. Dimensional quantities are as follows: α is the fluid’s coefficient of thermal expansivity; g is gravitational acceleration; ∆T is the temperature drop across the fluid layer; L is the depth of the fluid layer; ν is the fluid’s viscous diffusivity; κ is the thermal diffusivity of the fluid; Ω is the angular rotation rate; q is mean heat flux; and k is the fluid’s thermal conductivity. mass, and internal energy (heat) (e.g., Chandrasekhar 1953): ∂u 1 + u u +2Ω u = p + αgT + ν 2u ; (1.1) ∂t · ∇ × −ρ0 ∇ ∇ u = 0 ; (1.2) ∇ · ∂T + u T = κ 2T ; (1.3) ∂t · ∇ ∇ respectively, where u, p, and T are velocity, modified pressure, and temperature fields, respectively, ρ0 is the fluid’s mean density, and ν and κ are the viscous and thermal diffusivity of the fluid. There are seven global input parameters in the governing equations: Ω,ρ0, α, g, ∆T, ν, and κ; which contain four fundamental dimensions: time, mass, length, and temperature. By Buckingham’s Π-theorem, the system is therefore defined by three independent prognostic parameters (Barenblatt 2003). Although there are an infinite number of ways to combine these dimensional quantities into non-dimensional parameters, the fundamental control parameters are usually taken to be Ra, E, and P r, which are defined in table 1. The buoyancy forcing is characterized by the Rayleigh number, Ra. The period of rotation is characterized by the Ekman number, E. The Prandtl number, P r, characterizes the fluid itself as a ratio of viscous to thermal diffusivities. Rotating convection is generally found to occur in one of two basic regimes, geostrophic or weakly rotating, typically as evidenced by heat transfer behavior (e.g., Schmitz & Tilgner 2010). Heat transfer by geostrophic convection is suppressed relative to that by convection without rotation (e.g., Rossby 1969; Liu & Ecke 2009), and has been argued to scale as Nu Ra3E4 using marginal boundary layer stability analysis (King et al. 2012). Heat transfer∼ by weakly rotating convection is observed to scale similarly to that by convection without rotation (e.g., Liu & Ecke 1997; Niemela et al. 2010). In the range of Ra and P r accessed in this study, non-rotating convective heat transfer scales as Nu Ra2/7 (e.g., Chilla et al. 1993). As Ra increases, though, RBC studies observe that this∼ scaling exponent approaches the classical 1/3 value (e.g., Niemela & Sreenivasan 2006) predicted by the scaling analysis of ?. Recent work has argued that the transition between the geostrophic and weakly rotating heat transfer regimes is related to the relative thicknesses of the thermal and Ekman Page 3 of 21

Rotating convection scalings 3

Regime Geostrophic Weakly Rotating Non-rotating

Criterion RaE3/2 < 10 RaE3/2 > 10 E ≈∞

3 2/7 2/7 Heat Transfer Nu =(Ra/Rac) Nu = 0.16Ra Nu = 0.16Ra

Table 2. A summary of the results of King et al. (2012). The critical Rayleigh −4/3 number for rotating convection is Rac ≈ 7.6E (Chandrasekhar 1953), such that 3 3 4 Nu = (Ra/Rac) ≈ 0.0023Ra E . We stress that although the geostrophic scaling law follows from boundary layer stability analysis, the weakly and non-rotating 2/7 law is empirical, and probably a function of Ra and Pr (e.g. Ahlers et al. 2009).

boundary layers (King et al. 2009, 2010). Geostrophic convection, they argue, should occur when the Ekman boundary layer is thinner than the thermal boundary layer. When instead the thermal boundary layer is thinner, the influence of rotation should be weak. The two boundary layers are observed to cross when RaE3/2 10 in numerical simulations of rotating convection. Transitions between the two heat≈ transfer regimes are observed to occur near this boundary layer transition in both numerical simulations and laboratory experiments (King et al. 2012). These results are summarized in table 2. Here, we extend this framework to investigate the scaling behavior of five separate quantities in geostrophic (RaE3/2 . 10), weakly rotating (10 . RaE3/2 < ), and non- rotating (E = ) RBC simulations. The five quantities of interest are: visco∞us boundary ∞ layer thickness, δv (section 3); thermal boundary layer thickness, δT (section 4); prevailing thermal gradients, β (section 5); typical horizontal length scale of flow, ℓ (section 6); and typical flow speed, Pe (section 7).

2. Methods 2.1. Numerical Model Time evolution of the velocity, pressure, and temperature ﬁelds are calculated as numerical solutions of the governing equations (1.1-1.3) in non-dimensional form:

E ∂u′ + u′ u′ + ˆz u′ = P + RaE T ′ˆz ++E 2u′ (2.1) P r ∂t · ∇ × −∇ ∇ u′ = 0 (2.2) ′ ∇ · ∂T ′ ′ ′ + u T = 2T (2.3) ∂t · ∇ ∇ where u′, P , and T ′ are the dimensionless velocity, modified pressure, and temperature, respectively, and ˆz is the vertical unit vector. The fundamental length scale, L, is taken to be the height of the fluid layer. The fundamental time scale is taken to be a thermal diffusion time scale, L/κ2, where κ is the fluid’s thermal diffusivity. The temperature field is normalized by the temperature drop imposed across the layer, ∆T . The computational domain is a cartesian box with periodic sidewalls (in order to approximate an infinite plane layer); rigid, no-slip top and bottom boundaries; and diam- eter to height aspect ratio, 1 Γ 4. Temporal discretization is accomplished through a second-order, semi-implicit≤ Adams≤ Bashforth backward differentiation time stepping. Page 4 of 21

4 E. M. King, S. Stellmach, and B. Buﬀett Mean Horizontal Flow Speed 800 600 400 200 0 L

3L/4

U L/2 Trms T H Height

L/4

0 0 ΔΔΔT/4 T/2 3 T/4Δ T Temperature

′ Figure 1. An example of profiles of mean temperature (hT iH , solid line), temperature vari- ′ ′ ance (Trms, solid-filled curve), and mean horizontal flow speed (UH , stripe-filled curve), from a non-rotating convection simulation with Pr = 1 and Ra = 7 × 105. The horizontal dotted and dashed lines depict the calculated thickness of the thermal and viscous boundary layers, respectively.

Fourier series are used for spatial discretization in the horizontal direction, while Cheby- shev polynomials are used in the vertical direction in order to better resolve the thin fluid boundary layers. Ranges of input parameters accessed are given in Table 1. For further details on the numerical method and validation, please see Stellmach & Hansen (2008) and King et al. (2012). 2.2. Measurement Technique We define the following nomenclature for averaging: an overbar, , represents time averaging; angled brackets represent spatial averaging over the entire· · computational · domain, , and over horizontal planes, H . h· ·Viscous ·i boundary layer thicknessesh· · ·i are calculated as the mean vertical distances of the first local maximum of the horizontally averaged velocity variance above (below) the bottom (top) domain boundary. That is, a vertical profile for the mean magnitude of the horizontal velocity is acquired:

U ′ (z) u′ u′ 1/2; (2.4) H ≡ h H · H iH and the local peak of this profile nearest each of the top and bottom boundaries represent the edge of the top and bottom viscous boundary layers (Belmonte et al. 1994). An example profile is shown in Figure 1. The top and bottom layer thicknesses are averaged, giving the mean viscous boundary layer thickness, δv. Thermal boundary layer thicknesses are similarly calculated, using vertical profiles for temporal thermal variance, T ′ (z) T ′ T ′ . (2.5) rms ≡ h − h iH iH Page 5 of 21

Rotating convection scalings 5 4 10 Ekman Number

3 10 10 -3 10 -4 10 -5 2 -6 10 10 Prandtl Number

Peclet Number 1 1 10 7 100

0 10 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 Rayleigh Number Figure 2. Peclet number versus Rayleigh number, with Ekman numbers color-coded (non-rotating convection data are black), and Prandtl numbers are indicated by symbol size. This ﬁgure illustrates the parameter range accessed by numerical simulations.

Again, local peaks in this temperature fluctuation profile give estimates for the locations of the edges of the top and bottom thermal boundary layers, which are averaged to calculate δT . An example of these profiles is shown in Figure 1. The typical horizontal length scale of the flow, ℓ, is defined in the following way. Let ′ ′∗ (k,m) = [u(k,m)u(k,m)] be the dimensionless kinetic energy contained in horizontal modesE k and m. The dominant horizontal scale can be determined as

(kmax,mmax) 2π (k=1,m=1) (k,m) ℓ  E , (2.6) ≡ (kmaxP,mmax) (k,m)√k2 + m2 (k=1,m=1) E P  essentially calculating the mean wavelength of flow. The Nusselt number characterizes the heat transport as Nu u′ T ′ ∂T ′/∂z , (2.7) ≡ h z − i and the Peclet number is defined as the typical amplitude of flow speed: Pe (u′ u′) 1/2. (2.8) ≡ h · i Peclet number calculations are shown in Figure 2 plotted versus Ra to illustrate the parameter range accessed in this study. Ekman numbers are denoted by symbol shape (color online), and Prandtl numbers are denoted by symbol size. Statistical f-tests are used to distinguish quality-of-fit between different scalings for identical data. An f-test compares residual variance of data normalized by the two different scaling laws in question, and tests this comparison against the null hypothesis that the two residuals populations have equal variance to within 5% significance. That is, the ratio of the residual variances from the two scalings is compared with 95% confi- Page 6 of 21

6 E. M. King, S. Stellmach, and B. Buﬀett 0.3 a b −1 10 /L /L v v δ

δ −2 0.1 10

−3 0.05 10 0 1 2 3 −6 −5 −4 −3 10 10 10 10 10 10 10 10 Reynolds Number Ekman Number Figure 3. Symbols have the same meaning as in Figure 2. a) Viscous boundary layer thickness versus the Reynolds number, Re = Pe/Pr, for simulations without rotation (E = ∞). The −1/2 solid line illustrates the slope of the predicted scaling δv ∼ Re L. The dashed line shows −1/4 δv = 0.34Re L. b) viscous boundary layer thickness versus the Ekman number, for rotating convection simulations. Symbols have the same meaning as in Figure 2. The solid black line 1/2 shows δv = 3E L. dence bounds from an f-distribution with the same degrees of freedom as these residual populations (Snedecor & Cochran 1980).

3. Viscous boundary layers, δv 3.1. Non-rotating Viscous boundary layers in turbulent ﬂows are usually assumed to be of the Blasius type, derived by balancing the viscous term, which becomes important near the boundary, with the inertial term that is thought to dominate in the bulk (e.g., Kundu 1990). Matching these terms at the edge of the boundary layer gives: u u ν 2u U 2/L νU/δ2. (3.1) · ∇ ∼ ∇ → ∼ v The viscous boundary layer thickness, δv, should then scale as

2 −1/2 δv ν/U L = Re L, (3.2) ∼ p where Re = Pe/Pr is the Reynolds number. This boundary layer scaling represents the distance over which viscosity and acceleration by Reynolds stresses act on similar time scales. Figure 3a shows the thicknesses of the viscous boundary layer versus the Reynolds number from non-rotating convection simulations. The solid line shows the slope of the predicted scaling (3.2). The boundary layers within the simulations, however, are less strongly dependent on Re. A best ﬁt power law regression to these data with Re > 10 gives − ± δ =0.34( 0.05)Re 0.25( 0.03)L. (3.3) v ± This empirical scaling law is shown as the dashed line in Figure 3a, and agrees with the data to within 5% for Re > 10. It is not currently clear why the non-rotating viscous boundary layer scales weakly with Re, but this result is consistent with the extensive experimental study of Lam et al. (2002), and the numerical simulations of Breuer et al. (2004). We revisit this issue in the discussion section. Page 7 of 21

Rotating convection scalings 7 3.2. Rotating The viscous boundary layer in rotating ﬂows is known as an Ekman boundary layer, named after its discoverer V. W. Ekman (1905). The scaling behavior for the thickness of the Ekman layer comes from a balance between the Coriolis term and the viscous term near the top and bottom domain boundaries: 2Ω u ν 2u 2UΩ νU/δ2. (3.4) × ∼ ∇ → ∼ v The viscous boundary layer thickness, δv, should then scale as

2 1/2 δv ν/2ΩL = E L (3.5) ∼ p in rotating convection. This boundary layer scaling represents the distance over which viscosity acts in the course of one rotation period. Figure 3b shows calculations of the thickness of the viscous boundary layer versus E for 1/2 rotating convection. Our most rapidly rotating data appear to conform to a δv/L E (solid line). Indeed, we have twenty-six data points for rapidly rotating (E ∝10−4) geostrophic (RaE3/2 < 10) convection, and a least-squares power law regression≤ to these 0.49(±0.01) data yields δv/L =2.7( 0.3) E . To within the uncertainty, this scaling can be written as ± 1/2 δv/L =3E , (3.6) which is shown as the solid line in Figure 3b. This scaling law ﬁts the data to within 25% on average for all rotating cases (E 10−3), and to within 8% on average for E 10−4. ≤ ≤

4. Thermal boundary layers, δT 4.1. Non-rotating Turbulent convection, in the absence of constraining influences such as rotation, tends to mix the bulk fluid. Temperature within the interior becomes uniform as Ra is increased. The imposed temperature drop across the fluid layer is therefore accomplished almost entirely within the thermal boundary layers. The Nusselt number, Nu, is defined as the ratio of total heat transport to that by conduction alone. Conductive heat transport across the layer is given by qcond = k∆T/L. The total amount of heat transported by the fluid can be determined, in this idealized case, by the thickness of the boundary layer. Within the thermal boundary layer, heat transport is nearly entirely conductive, and so qtotal k∆T/2δT . Thus, for turbulent, non-rotating convection, we expect (e.g., Spiegel 1971)≈ 1 δ Nu−1L. (4.1) T ≈ 2 Figure 4 shows the thermal boundary layer thicknesses plotted versus Nu (non-rotating convection data are shown as circles). The solid line shows the predicted scaling (4.1), which fits the non-rotating data to within 15% for all cases, and to within 5% for Nu> 5 (Ra > 105).

4.2. Rotating In figure 4, we observe that the thermal boundary layer thickness in rotating convection is less well described by (4.1) than for convection without rotation. The quality of fit of the thermal boundary layer thickness calculations to usual Nu−1 scaling is especially poor within the geostrophic regime. To see this, we plot in figure 5a (δT /L)Nu versus the parameter, RaE3/2, which was previously defined to identify the transition between Page 8 of 21

8 E. M. King, S. Stellmach, and B. Buﬀett 0 10

−1

/L 10 T δ

−2 10 1 10 60 Nusselt Number

Figure 4. Thermal boundary layer thickness versus the Nusselt number for convection with and without rotation. Symbols have the same meaning as in ﬁgure 2. The solid line shows the slope of the predicted scaling (4.1), δT = L/(2Nu).

geostrophic and weakly rotating regimes. The solid horizontal line shows the predicted scaling for well-mixed turbulence (4.1), and the dotted horizontal lines indicate the standard deviation of non-rotating convection data about this prediction. In the geostrophic regime (RaE3/2 < 10) the misfit between thermal boundary layer thickness data and the classical prediction (4.1) (solid horizontal line) is 52% on average, and can be as high as 170%. For weakly rotating convection (RaE3/2 > 10), however, the thermal boundary layer thickness data are fit by (4.1) with an average error of 13% with a peak misfit of 31%. The poor agreement between (4.1) and geostrophic convection data is likely due to the breakdown of the assumption that the interior fluid is well mixed when a strong rotational influence is present. This affect of the Coriolis force has been previously observed in both experiments (e.g., Boubnov & Golitsyn 1990) and simulations (e.g., Julien et al. 1996). King et al. (2010) show that thermal mixing in geodynamo models can be subdued by rotation for increasingly higher Nu as E is decreased. They argue that as the Ekman layer becomes thinner than the thermal boundary layer, the interior fluid maintains significant thermal gradients, invalidating the classical thermal boundary layer thickness scaling (4.1). In figure 5b, we plot the relative thicknesses of the thermal and Ekman boundary layers, δT /δv, calculated for all rotating convection simulations versus our predicted transition 3/2 RaE . The boundary layers swap relative positions where δT /δv crosses unity, which occurs somewhere in the range 7 < RaE3/2 < 21 for all Ekman and Prandtl numbers considered. We approximate this transition as RaE3/2 10, shown as the dashed vertical line in figure 5. ≈ Page 9 of 21

Rotating convection scalings 9 1 2 a 10 b 0.8

0.7 1 10 0.6 v δ

0.5 / T δ /L Nu

T 0.4 0 δ 10 0.3

0.2 −1 0.1 10 0 1 2 3 0 1 2 3 10 10 10 10 10 10 10 10 RaE 3/ 2 RaE 3/2

Figure 5. a) Thermal boundary layer thickness normalized by the predicted scaling Nu−1, versus the transition parameter, RaE3/2. The solid horizontal line indicates the predicted scaling (4.1), and the dotted horizontal lines indicate the standard deviation of non-rotating convection data about (4.1) b) The ratio of thermal boundary layer thickness to viscous boundary layer thickness versus RaE3/2 for rotating convection. The dashed vertical lines in both panels indicate the approximate location of the boundary layer crossing. Symbols have the same meaning as in ﬁgure 2.

L L a b

3L/4 3L/4 E = ∞ − P r = 1 E = 10 3 P r = 7 −4 E = 10 P r = 100 −5 L/2 L/2 E E ∞ = 10 Height = Ra = 7 × 10 5 Height P r = 1 Nu ≈ 8

L/4 L/4

0 0 0 ΔΔT/4 T/2 3Δ T/4Δ T 0 ΔΔT/4 T/2 3Δ T/4Δ T Temperature Temperature

Figure 6. Mean temperature profiles for a) non-rotating and b) rotating convection. a) Tem- perature profiles for non-rotating convection with Ra = 7 × 105 and Pr = 1, Pr = 7, and Pr = 100. Thermal overshoot is observed as the Prandtl number is increased. b) Temperature profiles for Pr = 1 and: E = ∞ (non-rotating) and Ra = 7 × 105, solid curve, as in panel a; − − E = 10 3 and Ra = 7 × 105, dashed curve; E = 10 4 and Ra = 4 × 106, dash-dotted curve; and − E = 10 5 and Ra = 7 × 107, dotted curve. All four cases have 7.5

5. Interior temperature gradients, β The scaling relationship between heat flux and the thermal boundary layer thickness (4.1) hinges upon the assumption that the interior fluid is well mixed. Figure 6 shows mean temperature profiles for convection simulations with and without rotation. We observe that prevailing temperature gradients vary with P r for a given Ra in non-rotating convection (panel a), and with E for a given P r and Nu. In order to analyze these profiles more systematically, we quantify the degree of thermal mixing in the bulk fluid by calculating the mean vertical temperature gradient at mid-depth, which is non-dimensionalized Page 10 of 21

10 E. M. King, S. Stellmach, and B. Buﬀett 0.3 −0.001 a b 0.25

0.2 −0.01

0.15 β β

0.1 −0.1

0.05

0 −1 3 4 5 6 7 8 −2 −1 0 10 10 10 10 10 10 10 10 10 Rayleigh Number δv/δ T −0.2 0 d c

−0.2

−0.4 β β −0.5 −0.6

−0.8

−1 1 2 3 −1 10 10 10 7 10 15 25 RaE 4/ 3 RaE 4/ 3 Figure 7. Interior temperature gradients (5.1) for non-rotating (a) and rotating (b-d) convection. Symbols have the same meaning as in figure 2. a) Temperature gradients β, plotted against Ra for non-rotating convection. b) Temperature gradients calculated from rotating convection simulations plotted versus the ratio between Ekman and thermal boundary layers. The dashed vertical line illustrates the boundary layer transition (see figure 5), and the dotted horizontal line show the mean magnitude of β from the non-rotating simulations (panel a). c) Interior temperature gradients calculated from rotating convection simulations plotted versus supercriticality, RaE4/3. The solid line shows β = 6.6RaE4/3. d) Shows an expanded view of the same data as in panel c. by the imposed, global gradient, ∂T ′ β (z = L/2) (L/∆T ). (5.1) ≡ h ∂z iH 5.1. Non-rotating Figure 7a shows calculations of the interior temperature gradients, β, plotted versus the Rayliegh number for the non-rotating simulations. Two notable observations arise. First, convection with high Rayleigh numbers (Ra > 106) and/or low Prandtl numbers (P r = 1) are generally well mixed thermally. Second, the prevailing temperature gradients are positive, whereas the imposed global gradient is negative. This overshoot of the background gradient can be seen explicitly in the thermal profiles plotted in figure 6a. Thermal overshoot has been observed previously for convection with low Re, and is attributed to convective flows dominated by long-lived thermal plumes (e.g., Olson & Corcos 1980). These plumes traverse the layer as a quasi-Stokes flow, and collect along Page 11 of 21

Rotating convection scalings 11 the opposite boundary where their temperature anomalies produce local peaks in the mean temperature proﬁle.

5.2. Rotating We observed in section 4.2 that rotation upsets the assumptions necessary to reach (4.1) (see figure 5). Previous studies have shown that the Coriolis force is capable of inhibiting turbulent mixing in general (Taylor 1921), in convection experiments (Boubnov & Golitsyn 1990; Fernando et al. 1991) and simulations (Julien et al. 1996; Sprague et al. 2006), and in convective dynamo simulations (King et al. 2010). Understanding the interplay between rotation and thermal mixing is essential for adequate parameterization of small scale convection in the oceans and core of Earth and other planets. King et al. (2010) argue that the inhibition of thermal mixing for convection systems occurs when the Ekman boundary layer is thinner than the thermal boundary layer, following arguments for Nu scaling behavior (King et al. 2009). Our current results, shown in figure 7b, are consistent with this suggestion. Figure 7b shows interior temperature gradient calculations, β, are plotted versus the ratio between the measured thicknesses of the boundary layers. The dashed vertical line indicates the boundary layer transition, where δv = δT . The dotted horizontal line shows the mean magnitude of the mid-plane gradient for the non-rotating simulations for comparison. We observe that rotation sup- presses thermal mixing when the Ekman layer is the thinner boundary layer. Figures 7c,d show the interior temperature gradient calculations plotted versus supercriticality, RaE4/3 (onset of convection for the present range of E and P r should occur near RaE4/3 = 7 (Chandrasekhar 1953)). Sprague et al. (2006) observe that, near onset for an asymptotically reduced system, temperature gradients scale with the inverse of supercriticality. Our results are consistent with this observation. A best fit power law scaling for the rotating convection simulations that have δE < 0.5δT yields β = 6.2( 2.3)(RaE4/3)−0.98(±0.14). On average for these rapidly rotating simulations, β/(RaE− 4/3±)= 6.6( 1.2), which is shown as the solid line in figures 7c,d, and fits these data to within− 6% on± average.

6. Bulk length scales, ℓ 6.1. Non-rotating It is often observed in experimental studies of Rayleigh-Bénard convection that thermal turbulence organizes into large-scale circulation patterns (also known as mean thermal winds or flywheel convection patterns) (Ahlers et al. 2009). Such aggregate patterns of thermal turbulence result from the presence of rigid sidewalls, which help to organize smaller scale turbulence into a sweeping cell that follows the edges of the container (Xi et al. 2004). In the present simulations, however, the absence of rigid sidewalls eliminates this selection process, and large scale circulation patterns are not observed. The length scales for flow in our simulations are instead naturally selected, and should be the typical separation of thermal plumes. Figure 8 illustrates this plume separation, ℓ. We develop a scaling for ℓ by considering the timescale for flow along the boundary. If fluid flows along the boundary with typical speed ub, then the travel time between plumes is τ ℓ/ub. As this flow travels along the boundary, a thermal front will propagate toward the∼ interior. This diffusive growth is illustrated in figure 8b as thermal profiles within the boundary layer flow. The boundary flow should become unstable when this thermal anomaly becomes roughly as thick as the mean thermal boundary layer. Thus, we can relate the boundary flow and plume Page 12 of 21

12 E. M. King, S. Stellmach, and B. Buﬀett a rp rp

δT ub ub

T δT ub

Figure 8. A schematic of the plume separation length scale. The typical separation between plumes is denoted by ℓ, the thermal boundary layer thickness by δT , typical plume speed by up, the typical plume width by rp, and the flow along the boundary by ub. development timescale as: τ ℓ/u δ2 /κ. (6.1) ∼ b ∼ T Finally, we can determine how the boundary layer flow speed should scale with Pe in order to estimate the characteristic length scale ℓ. Figure 8a illustrates the three- dimensional geometry of plumes and boundary layer. The plumes have radius rp and travel vertically with speed up. Boundary layer flow spreads (converges) horizontally along the boundary away from impinging plumes (toward departing plumes) with speed ub. Conservation of mass flux dictates that u πr2 = u πℓδ . (6.2) p · p b · T Assuming that u Pe(κ/L), and that the plume width scales with thermal boundary p ∼ layer thickness, rp δT , we can combine (6.1) and (6.2) to estimate the natural plume spacing in non-rotating∼ convection:

δ 3/2 ℓ T Pe1/2 L. (6.3) ∼ L A scaling law for plume spacing in high P r convection was developed by Parmentier & Sotin (2000) following similar reasoning. Figure 9a shows calculations of the characteristic horizontal length scale ℓ versus the Page 13 of 21

Rotating convection scalings 13 0.7 0.3 a b

0.1

0.3 L / L ℓ / ℓ 0.03

0.01

0.1 0.1 1 −6 −5 −4 −3 1/ 2 3/ 2 10 10 10 10 P e (δT /L ) Ekman Number Figure 9. Characteristic length scale for non-rotating (b) and rotating (b) convection. Sym- 1/2 3/2 bols have the same meaning as in ﬁgure 2. a)ℓ/L versus Pe (δT /L) (from (6.3)), from 1/2 3/2 non-rotating convection simulations. The solid line shows ℓ/L = 0.8Pe (δT /L) . b) Typical horizontal length scale versus E for rotating convection. The solid line shows ℓ = E1/3L.

1/2 3/2 this scaling (6.3) from non-rotating convection simulations. The scaling ℓ/L =0.8Pe (δT /L) is shown as a solid line, and describes the data to within 25% on average. An alternative, but complementary estimate of interior length scales in non-rotating convection can be determined by heat flux considerations in the bulk. If plumes typically have radius rp and average spacing ℓ (see figure 8), then the fraction of a horizontal 2 cross-section through the bulk fluid occupied by plumes is π(rp/ℓ) . Mean convective ′ ′ heat flux, Nu 1= u T , may scale with the typical plume speed up Pe, multiplied by this plume− densityh fraction.i If we again assume that the typical width∼ of the plumes scales with the thermal boundary layer thickness, r δ , then p ∼ T Pe 1/2 ℓ δ . (6.4) ∼ T Nu 1 − If we further assume that the interior fluid is well mixed such that δT 1/Nu and Nu 1 Nu, then we recover the scaling derived by considering boundar∼y layer flow (6.3).− ≈

6.2. Rotating Rotating RBC cannot be purely geostrophic, since mean convective heat transport re- quires vertical flow, which cannot be z-invariant in a layer of finite thickness. Convection in the presence of strong Coriolis forces must break the Taylor-Proudman (TP) theorem. For fluids with P r & 1, it is usually assumed that the TP theorem is broken by viscous forces acting on small horizontal length scales. The characteristic length scale for kinetic energy should then occur at the scale on which viscosity can balance the Coriolis term in the horizontal direction. Taking the curl of the momentum equation and retaining only these two terms, we have ∂u 2Ω ν 2ω, (6.5) ∂z ∼ ∇ where ω = u is the fluid vorticity. The next-order terms in the TP theorem tells us that ∂/∂z ∇×1/L (e.g., Greenspan 1968; King & Aurnou 2012), so the lhs of 6.5 scales as ΩU/L. The∼rhs of 6.5, in contrast, is dominated by horizontal gradients, 2ω U/ℓ3. This interior balance therefore predicts vortical flow with characteristic horizontal∇ ∼ length Page 14 of 21

14 E. M. King, S. Stellmach, and B. Buffett scale ℓ/L E1/3. (6.6) ∼ This characteristic scale also corresponds to the critical length scale for the onset of convection in linear analysis of rapidly rotating convection (Chandrasekhar 1953). Figure 9b shows calculations of typical horizontal length scale ℓ/L for rotating convection plotted versus the Ekman number. A best fit power law scaling yields: ℓ/L =1.09( 0.14) E0.34(±0.01); (6.7) ± for the geostrophic cases (RaE3/2 < 10), in agreement with the scaling prediction. The solid line in figure 9b shows ℓ/L = E1/3, which fits the rotating convection data to within an average of 3%, and fits within 1% on average for data in the geostrophic regime (RaE3/2 < 10).

7. Flow speeds, P e 7.1. Non-rotating Much work has been done to relate the typical speed of convective motions (Pe) with the strength of driving (Ra) in both laboratory experiments and numerical simulations (e.g., Ahlers et al. 2009). A useful tool for such scaling is the exact balance between mean global production and dissipation of kinetic energy: κ2 (Nu 1)Ra = ( u)2 , (7.1) h4 − h ∇ i which follows from taking the global average of u (1.1) (e.g., Shraiman & Siggia 1990). · 2 2 2 2 The rightmost term can be approximated as either U /ℓ or U /δv, by assuming that the relevant length scale for dissipation is either the characteristic interior length scale, ℓ, or the boundary layer thickness, δv, respectively (Grossmann & Lohse 2000). Unfortunately, in our simulations we find no regimes in which dissipation is dominated by either the bulk or the boundary layers, in agreement with the results of Calzavarini et al. (2005). Substitution of each of these approximations into (7.1) produces scalings for convective speed based on dissipation in the bulk, Pe∗ (Nu 1)1/2 Ra1/2 (ℓ/L), (7.2) int ≡ − and dissipation in the boundary layers, Pe∗ (Nu 1)1/2 Ra1/2 (δ /L), (7.3) bl ≡ − v Figures 10a and 10b show Peclet number calculations plotted versus these dissipation ∗ 1.04(±0.04) integral scalings. A best fit power law regression to each gives Pe =0.28( 0.06) Peint ∗ 0.99(±0.03) ± for the bulk dissipation estimate, and Pe =0.59( 0.09) Pebl for the boundary ± ∗ layer dissipation estimate. Imposing linear fits, we get Pe/Peint = 0.36( 0.05) and ∗ ± Pe/Pebl = 0.56( 0.07), which are shown in figures 10a and 10b, respectively. An f-test reveals that these± fits have statistically indistinguishable differences. These scalings suggest that convective flow may strike a balance between dissipation in the bulk and boundary layers. Such a balance can also be seen in the time-averaged heat equation, ∂ ∂2 uT ′ = κ T , (7.4) ∂z ∂z2 where T ′ is a typical temperature fluctuation. Assuming an isothermal interior and well- defined thermal boundary layers, we can scale this balance as U∆T/L κ∆T/δ2 . Using ≈ T Page 15 of 21

Rotating convection scalings 15 4 4 10 10 a non-rotatingb non-rotating

3 3 10 10

2 2 10 10

1 1 Peclet10 Number Peclet10 Number

0 0 10 10 1 2 3 4 1 2 3 4 10 10 10 10 10 10 10 10 ∗ ∗ P e int P e bl 4 4 10 10 c non-rotating d rotating

3 10 3 10

2 10 2 10

1 Peclet Number Peclet10 Number 1 10

0 10 1 2 3 4 1 3 10 30 10 10 10 10 Nusselt Number ∗ P e int Figure 10. Peclet number scalings for non-rotating (a-c) and rotaing (d) convection. Sym- bols have the same meaning as in ﬁgure 2. a) Peclet number versus the interior dissi- ∗ pation scaling, Peint = pRa(Nu − 1)(ℓ/L), for non-rotating convection. The solid line ∗ shows the scaling Pe = 0.36Peint. b) Peclet number versus the boundary layer dissipa- ∗ tion scaling, Pe = pRa(Nu − 1)(δv/L), for non-rotating convection. The solid line shows ∗ bl Pe = 0.56Pebl. c) Peclet number versus the Nusselt number for non-rotating convection. The solid line shows Pe = 2.5Nu2. d) Peclet number versus the interior dissipation scaling, ∗ ∗ Pebl = pRa(Nu − 1)(δv/L), for rotating convection. The solid line shows Pe = 0.65Peint.

(4.1), this balance gives a Peclet number scaling in terms of the Nusselt number alone, Pe 4Nu2. (7.5) ≈ Figure 10c shows Pe plotted versus Nu. A best fit power law regression yields Pe = 2.0( 0.3) Nu2.11(±0.07). When Nu > 4, for which (4.1) was shown to hold, our data agree± better with (7.5), as Pe =3.1( 0.8) Nu1.93(±0.11). Imposing a scaling of the form (7.5) gives Pe/Nu2 =2.5( 0.4), which± gives an average misfit of 21% for all non-rotating data and 11% for Nu> 4,± and is shown as the solid line in figure 10c. We caution, though that, since this scaling is derived solely from the heat equation, it is unlikely to hold for very large Reynolds numbers, when inertia is expected to play a dominant role. And we do observe that the quality of the fit declines for decreasing P r and increasing Ra.

7.2. Rotating As the rate of rotation increases (decreasing E), we ﬁnd that viscous dissipation is increasingly dominated by its interior (bulk) contribution. We therefore expect that ﬂow Page 16 of 21

16 E. M. King, S. Stellmach, and B. Buffett speeds will follow the interior dissipation scaling (7.2). Figure 10d shows calculations of convective flow speed Pe plotted versus the interior dissipation scaling (7.2). A best fit ∗ 0.99(±0.04) power law regression yields Pe = 0.62( 0.14) Peint for all rotating convection ± ∗ simulations, in agreement with the prediction (7.6). The average value of Pe/Peint is 0.65, which is shown as the solid line in figure 10d, and fits all of the rotating convection data with an average error of 36%. The Coriolis force can do no work, and so only enters the energetic flow speed scaling (7.2) through its influence on length scales. Fortunately, the characteristic length scales of rapidly rotating convection follow well known scaling relationships in terms of E alone (6.6). Substituting (6.6) into (7.2) produces a prediction for flow speed based on the input parameter E rather than calculations of length scales: Pe Pe∗ (Nu 1)1/2 Ra1/2E1/3. (7.6) ∼ V AC ≡ − Here, the subscript ‘V AC’ stands for visco-Archimedean-Coriolis, for the triple force balance on which it is based. This scaling provides a similar quality-of-fit to the scaling using actual, calculated lengtch scales, and so is not plotted here. Best fit power law re- ∗ 0.94(±0.03) gressions yield Pe =0.9( 0.2) PeVAC for all rotating convection simulations, and ∗ 1.01(± ±0.01) 3/2 Pe = 0.66( 0.04) PeVAC for geostrophic simulations (RaE < 10), in agree- ± ∗ 3/2 ment with the prediction (7.6). The average value of Pe/PeV AC for RaE < 10 is ∗ 0.75, and Pe =0.75PeVAC fits all of the rotating convection data with an average error of 43%, and fits the geostrophic cases (RaE3/2 < 10) to within 8% on average.

8. Discussion Many of the scaling laws presented here can be combined to produce new scalings without reducing agreement with data. For example, combining (7.5) with (6.3) and (4.1) generates a scaling for the characteristic length scale in non-rotating as a function of the thermal boundary layer thickness alone, ℓ/L (δ /L)1/2. (8.1) ∼ T This scaling describes the data as well as (6.3): with a prefactor of 0.73, it fits the non- rotating data to within 16% on average (and quality-of-fit improves with increasing Ra). Furthermore, this scaling is identical to that observed by Parmentier & Sotin (2000) and Zhong (2005). Similarly, for the non-rotating viscous boundary layer thickness, we can combine (3.3), (7.5), and the classical heat transfer scaling, Nu Ra1/3, to produce: ∼ δ /L Ra−1/6P r1/4. (8.2) v ∼ We emphasize that this scaling is empirically based, since we offer no theoretical ex- planation for (3.3). This relationship is, however, in close agreement with the empirical scaling observed in Lam et al. (2002), whose non-rotating experiments span the ranges 106 . Ra . 1011 and 10−2 . P r . 103, as well as the numerical work of Breuer et al. (2004), who utilize a similar geometry to ours, but reach Re 2 104. We feel this agree- −1/4 ≈ × ment substantiates our finding that δv Re rather than the typical Blasius scaling, −1/2 ∼ δv Re . This difference is important as many theoretical treatments of the turbulent∼ convection problem assume the latter (e.g., Grossmann & Lohse 2000). We suggest that the Blasius scaling is not observed in these simulations and experiments because the boundary layer is active, rather than a passive response to bulk turbulence assumed in (3.1) and (3.2). Qiu and Xia measure viscous boundary layer thicknesses along both the Page 17 of 21

Rotating convection scalings 17 sidewall (1998a) and bottom (1998b) boundaries. They find that the sidewall boundary layer has Blasius-type Re dependence, while the bottom boundary layer thickness has a weaker dependence, scaling closer to (8.2). (For Boussinesq convection, we expect the same holds for the top boundary.) This supports the argument that the top and bottom boundary layers, through which heat is fluxed, are dynamically active and so have thicknesses that scale differently from the Blasius type. At the sidewall, however, which is roughly adiabatic, one might expect that the boundary layer is a passive response to bulk turbulence. The Blasius boundary layer scaling is also recovered in experiments where a strong mean wind is imposed by the tank geometry (Sun et al. 2008). Our simulations use periodic sidewall boundary conditions in order to eliminate the effects of sidewalls, which are absent in the original formulation of the Bénard problem. The results of the non-rotating convection simulations shown here suggest that the important length scales in RBC, δT , δv, and ℓ, are dynamically interdependent. Combining (3.3), (7.5), and (4.1), we get another scaling relationship for the viscous boundary layer in non-rotating convection, δ /L (δ /L)1/2P r1/4. (8.3) v ∼ T Scaling relationships (8.1) and (8.3) describe the connections between these length scales. Rotating convection is somewhat simpler in this regard, as the scaling behavior of δv and ℓ are well-behaved responses to particular balances between the Coriolis force and viscosity. The opposite is true for δT : in non-rotating convection the interior is effectively mixed such that (4.1) roughly holds. In rotating convection, the Coriolis force inhibits this mixing, permitting finite interior temperature gradients (figure 7), and complicating the usual simple thermal boundary layer controlled depiction of heat transfer. Of course, care must be taken when amalgamating or extrapolating the scalings presented here. For example, for non-rotating flow speeds, we can combine (4.1), (7.2), and (8.1) to get, in the limit of high Nu, Pe Ra1/2. (8.4) ∼ If, however, we instead use (7.5) and (4.1) with Nu Ra1/3, we arrive at ∼ Pe Ra2/3. (8.5) ∼ We emphasize in section 7.1 that (7.5), and therefore (8.5), is likely to hold only for low Re (high P r and/or low Ra). We expect (8.4) should, in contrast, hold for high Re. Experiments (Lam et al. 2002) and simulations (Silano et al. 2010) exploring broad ranges of P r support this expectation. If we maintain the Nu dependence in the Pe scaling for non-rotating convection, we find better convergence. Using (4.1) with (7.2) and (8.1) (or, equivalently, with (7.3) and (8.3) and neglecting the P r-dependence in the latter), we arrive at Pe (Ra Ra/Nu)1/2. (8.6) ∼ − This scaling law is shown in figure 11a with a prefactor of 0.18, which fits the non-rotating convection data to within 14% on average. Previous work has investigated transitions in heat transfer behavior for rotating convection by comparing Nu from rotating and non-rotating experiments and simulations. We can do the same for flow speeds, Pe. First, we must characterize the non-rotating Pe behavior in a simple way. For this, we take a best fit power law regression of Pe with Ra for E = and Ra > 105, which yields Pe =0.14( 0.1)Ra0.52(±0.04). Figure 11b shows flow speeds∞ from rotating convection simulations normalize± d by this scaling, PeRa−0.52, plotted versus the transition parameter, RaE3/2. We observe a change in behavior near Page 18 of 21

18 E. M. King, S. Stellmach, and B. Buﬀett

4 0 10 10 a b

3 10 −1 10 52 . 0

2 − 10

P eRa −2 10 1 Peclet10 Number

0 10 −3 2 3 4 5 6 7 8 10 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 Ra − Ra /Nu RaE 3/2

Figure 11. a) Peclet number plotted versus Ra−Ra/Nu from (8.6) for non-rotating convection. The solid line depicts Pe = 0.2(Ra−Ra/Nu)1/2. b)Transition between rotating and non-rotating flow speed behaviors. Peclet numbers from rotating convection simulations are normalized by the non-rotating fit Ra0.52 and plotted agains the boundary layer transition parameter. The solid horizontal line shows the non-rotating fit, Pe = 0.14Ra0.52. The dotted vertical line indicates the boundary layer transition from figure 5b. the boundary layer transition RaE3/2 10 (see figure 5b) between suppressed flow speeds and flow speeds that conform to the≈ non-rotating behavior indicated by the horizontal line. In agreement with the heat transfer transitions observed in King et al. (2009, 2012), this flow speed transition is not well described by the convective Rossby number, 2 Roc = RaE /P r. Thus, our simulations indicate that the scaling behavior of both global heatp transfer and mean flow speeds are governed by the relative thicknesses of the thermal and Ekman boundary layers. One perhaps surprising result of note is that our quantification of the influence of rotation does not agree with that by the oft-used Rossby number. Not to be confused with the convective Rossby number, the Rossby number, Ro = PeE/Pr, is used to characterize the relative importance of inertial and Coriolis forces. It is often assumed that fluid systems with low Rossby number (Ro < 1) should be geostrophic. We find instead that the the influence of rotation on the quantities of interest here is better described by the transition parameter RaE3/2, which characterizes the relative thicknesses of the thermal and Ekman boundary layers (figure 5b). In our simulations, we observe weakly rotating convection cases (RaE3/2 > 10) that have Rossby numbers as small as Ro . 10−3. Fur- thermore, cases with Ro < 1 are found even at our highest finite value of RaE3/2 = 7000. Our results therefore indicate that low Rossby numbers do not necessitate geostrophic convection.

9. Summary We investigate the scaling behavior of five quantities produced in rotating and non- rotating convection simulations in a cartesian box with periodic sidewalls, operating within the parameter ranges 10−6 E , 1 P r 100, and 103 . Ra . 109. The ≤ ≤ ∞ ≤ ≤ quantities of interest are the viscous boundary layer thickness, δv, thermal boundary layer thickness, δT , mid-layer mean temperature gradient, β, characteristic bulk length scale, ℓ, and mean flow speeds, Pe. Table 3 summarizes our results. We find three important convection regimes: geostrophic convection, RaE3/2 . 10, Page 19 of 21

Rotating convection scalings 19

Table 3. Summary of results. Regarding the weakly and non-rotating heat transfer scalings, we stress that the 2/7 law is empirical, and likely increases with Ra to a 1/3 law (e.g., Niemela & Sreenivasan 2006; Niemela et al. 2010). Page 20 of 21

20 E. M. King, S. Stellmach, and B. Buffett where flow speeds and heat transfer are suppressed by the Coriolis force; weakly rotating convection, 10 . RaE3/2 < , where flow speeds and heat transfer are not strongly ∞ affected by rotation, but δv and ℓ are still dictated by the Coriolis force; and non-rotating convection. The transition from geostrophic to weakly rotating convection occurs when the thermal boundary layer becomes thinner than the Ekman layer.

REFERENCES Ahlers, G, Grossmann, S & Lohse, D 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Reviews of Modern Physics 81, 503–537. Barenblatt, G I 2003 Scaling. Cambridge University Press. Belmonte, A, Tilgner, A & Libchaber, A 1994 Temperature and velocity boundary layers in turbulent convection. Physical Review E 50 (1), 269–280. Benard,´ H 1900 Etude´ expérimentale des courants de convection dans une nappe liquide.— régime permanent: tourbillons cellulaires. Journal de Physique Théorique et Appliquée 9, 513–524. Boubnov, B & Golitsyn, G 1990 Temperature and velocity-field regimes of convective motions in a rotating plane fluid layer. Journal of Fluid Mechanics 219, 215–239. Breuer, M, Wessling, S, Schmalzl, J & Hansen, U 2004 Effect of inertia in rayleigh-bénard convection. Physical Review E 69 (2), 026302. Calzavarini, E, Lohse, D, Toschi, F & Trippiccione, R 2005 Rayleigh and Prandtl number scaling in the bulk of Rayleigh–Bénard turbulence. Physics of Fluids 17, 055107. Chandrasekhar, S 1953 The instability of a layer of fluid heated below and subject to Coriolis forces. Proceedings of the Royal Society of London Series A 217 (1130), 306–327. Chilla, F, Ciliberto, S, Innocenti, C & Pampaloni, E 1993 Boundary layer and scaling properties in turbulent thermal convection. Il Nuovo Cimento D . Ekman, V 1905 On the influence of the Earth’s rotation on ocean currents. Ark. Mat. Astron. Fys. 2, 1–53. Fernando, H J S, Chen, RR & Boyer, D L 1991 Effects of rotation on convective turbulence. Journal of Fluid Mechanics 228, 513–547. Greenspan, H 1968 The Theory of Rotating Fluids. Cambridge University Press. Grossmann, S & Lohse, D 2000 Scaling in thermal convection: a unifying theory. Journal of Fluid Mechanics 407, 27–56. Julien, K, Legg, S, McWilliams, J & Werne, J 1996 Rapidly rotating turbulent Rayleigh– Bénard convection. Journal of Fluid Mechanics 322, 243–273. King, E, Soderlund, K, Christensen, U R, Wicht, J & Aurnou, J M 2010 Convective heat transfer in planetary dynamo models. Geochemistry Geophysics Geosystems 11 (6), Q06016. King, E, Stellmach, S, Noir, J, Hansen, U & Aurnou, J 2009 Boundary layer control of rotating convection systems. Nature 457 (7227), 301–304. King, E M & Aurnou, J M 2012 Thermal evidence for taylor columns in turbulent rotating Rayleigh-Bénard convection. Physical Review E 85, 016313. King, E M, Stellmach, S & Aurnou, J M 2012 Heat transfer by rapidly rotating Rayleigh– Bénard convection. Journal of Fluid Mechanics 691, 568–582. Kundu, P K 1990 Fluid Mechanics. Academic Press. Lam, Siu, Shang, Xiao-Dong, Zhou, Sheng-Qi & Xia, Ke-Qing 2002 Prandtl number dependence of the viscous boundary layer and the Reynolds numbers in Rayleigh-Bénard convection. Physical Review E 65 (6), 066306. Liu, YM & Ecke, RE 1997 Heat transport scaling in turbulent Rayleigh–Bénard convection: Effects of rotation and Prandtl number. Physical Review Letters 79 (12), 2257–2260. Liu, Y & Ecke, R 2009 Heat transport measurements in turbulent rotating Rayleigh-Bénard convection. Physical Review E 80 (3), 6314–6325. Niemela, J. J, Babuin, S & Sreenivasan, K. R 2010 Turbulent rotating convection at high Rayleigh and Taylor numbers. Journal of Fluid Mechanics 649, 509. Niemela, J. J & Sreenivasan, K. R 2006 Turbulent convection at high Rayleigh numbers and aspect ratio 4. Journal of Fluid Mechanics 557, 411–422. Page 21 of 21

Rotating convection scalings 21 Olson, P & Corcos, G M 1980 A boundary layer model for mantle convection with surface plates. Geophysical Journal of the Royal Astronomical Society 62, 195–219. Parmentier, E M & Sotin, C 2000 Three-dimensional numerical experiments on thermal convection in a very viscous fluid: Implications for the dynamics of a thermal boundary layer at high Rayleigh number. Physics of Fluids 12 (3), 609–617. Qiu, XL & XIa, K-Q 1998a Spatial structure of the viscous boundary layer in turbulent convection. Physical Review E 58 (5), 5816. Qiu, XL & XIa, K-Q 1998b Viscous boundary layers at the sidewall of a convection cell. Physical Review E 58 (1), 486–491. Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philosophical Magazine Series 6 32 (192), 529–546. Rossby, H 1969 A study of Bénard convection with and without rotation. Journal of Fluid Mechanics 36 (2), 309–335. Schmitz, S & Tilgner, A 2010 Transitions in turbulent rotating Rayleigh-Benard convection. Geophysical and Astrophysical Fluid Dynamics pp. 1–8. Shraiman, Boris I & Siggia, Eric D 1990 Heat transport in high-Rayleigh-number convection. Physical Review A 42 (6), 3650–3653. Silano, G, Sreenivasan, K. R & Verzicco, R 2010 Numerical simulations of Rayleigh– Bénard convection for Prandtl numbers between 101 and 104 and Rayleigh numbers between 105 and 109. Journal of Fluid Mechanics 662, 409–446. Snedecor, GW & Cochran, W G 1980 Statsitical Methods. Iowa State University Press. Spiegel, E 1971 Convection in stars. I. Basic Boussinesq convection. Annual Review of Astron- omy and Astrophysics 9, 323–352. Sprague, M, Julien, K, Knobloch, E & Werne, J 2006 Numerical simulation of an asymptotically reduced system for rotationally constrained convection. Journal of Fluid Mechanics 551, 141–174. Stellmach, S & Hansen, U 2008 An efficient spectral method for the simulation of dynamos in Cartesian geometry and its implementation on massively parallel computers. Geochemistry Geophysics Geosystems 9 (5). Sun, C, Cheung, YH & XIa, K-Q 2008 Experimental studies of the viscous boundary layer properties in turbulent Rayleigh–Bénard convection. Journal of Fluid Mechanics 605, 79– 113. Taylor, G I 1921 Experiments with rotating fluids. Proceedings of the Royal Society of London Series A 100 (703), 114–121. Xi, H, Lam, S & Xia, K 2004 From laminar plumes to organized flows: the onset of large-scale circulation in turbulent thermal convection. Journal of Fluid Mechanics 503, 47–56. Zhong, Shijie 2005 Dynamics of thermal plumes in three-dimensional isoviscous thermal convection. Geophysical Journal International 162 (1), 289–300.