# Reynolds number scaling and energy spectra in geostrophic convection

Matteo Madonia1  Andrés J. Aguirre Guzmán1  Herman J.H. Clercx1 and Rudie P.J.
Kunnen1<EMAIL_ADDRESS>1Fluids and Flows group, Department of Applied
Physics and J. M. Burgers Centre for Fluid Dynamics, Eindhoven University of
Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

###### Abstract

We report flow measurements in rotating Rayleigh–Bénard convection in the
rotationally-constrained geostrophic regime. We apply stereoscopic particle
image velocimetry to measure the three components of velocity in a horizontal
cross-section of a water-filled cylindrical convection vessel. At a constant,
small Ekman number $Ek=5\times 10^{-8}$ we vary the Rayleigh number $Ra$
between $10^{11}$ and $4\times 10^{12}$ to cover various subregimes observed
in geostrophic convection. We also include one nonrotating experiment. The
scaling of the velocity fluctuations (expressed as the Reynolds number $Re$)
is compared to theoretical relations expressing balances of
viscous–Archimedean–Coriolis (VAC) and Coriolis–inertial–Archimedean (CIA)
forces. Based on our results we cannot decide which balance is most applicable
here; both scaling relations match equally well. A comparison of the current
data with several other literature datasets indicates a convergence towards
diffusion-free scaling of velocity as $Ek$ decreases. However, the use of
confined domains leads at lower $Ra$ to prominent convection in the wall mode
near the sidewall. Kinetic energy spectra point at an overall flow
organisation into a quadrupolar vortex filling the cross-section. This
quadrupolar vortex is a quasi-two-dimensional feature as it only manifests in
energy spectra based on the horizontal velocity components. At larger $Ra$ the
spectra reveal the development of a scaling range with exponent close to
$-5/3$, the classical exponent for inertial-range scaling in three-dimensional
turbulence. The steeper $Re(Ra)$ scaling at low $Ek$ and development of a
scaling range in the energy spectra are distinct indicators that a fully
developed, diffusion-free turbulent flow state is approached, sketching clear
perspectives for further investigation.

##  1 Introduction

Buoyant convection and rotational influence through Coriolis acceleration are
two principal features of fluid flows encountered in geophysics and
astrophysics. A popular model system to study buoyant convective flows is
Rayleigh–Bénard convection (RBC): a horizontal layer of fluid sandwiched
between two parallel plates where the bottom plate is at a higher temperature
than the top plate. With the inclusion of background rotation about a vertical
axis, the problem of rotating Rayleigh–Bénard convection (RRBC) encompasses
the principal actors, convection and rotation, in a simple, mathematically
well-defined system amenable to experimental, numerical and analytical
investigation. The interplay of convection and rotation is often studied in
the RRBC context (Stevens et al., 2013a; Kunnen, 2021; Ecke & Shishkina,
2023), which in itself has so far presented a wide variety of possible flow
states without invoking additional complexities that may be encountered in
geo- and astrophysical flows such as magnetic fields, strong non-
Oberbeck–Boussinesq effects, spherical geometries, and more (e.g., Jones,
2011; Glatzmaier, 2014; Aurnou et al., 2015; Schumacher & Sreenivasan, 2020).

A property shared by the majority of geo- and astrophysical flows is that they
are characterised by large Reynolds numbers (ratio of inertial to viscous
forces) and at the same time small Rossby numbers (ratio of inertial to
Coriolis forces). We thus expect turbulent convective flow with a strong
rotational constraint. In recent RRBC investigations it was found that
rotation-dominated convection — named the geostrophic regime after the
dominant balance between Coriolis force and pressure gradient — displays an
interesting subdivision into various realisable flow phenomenologies, each
with specific scalings of descriptive statistical quantities like the
efficiency of convective heat transfer (e.g., Sprague et al. (2006); Julien et
al. (2012b); Stellmach et al. (2014); Ecke & Niemela (2014); Cheng et al.
(2015, 2020); Lu et al. (2021); Madonia et al. (2021)), which is recently
reviewed in Kunnen (2021) and Ecke & Shishkina (2023). Starting from onset of
convection, with increasing strength of buoyant forcing one expects to
encounter cellular convection, convective Taylor columns (CTCs), plumes and
geostrophic turbulence (GT), showing step by step a decreasing rotational
constraint (Sprague et al., 2006; Julien et al., 2012b; Kunnen, 2021; Aguirre
Guzmán et al., 2022). From our prior heat transfer and temperature
measurements (Cheng et al., 2020) we could identify an additional transitional
state, rotation-influenced turbulence (RIT), that requires further
description.

Getting to the geostrophic regime requires dedicated tools: large-scale
experimental setups (e.g., Ecke & Niemela, 2014; Cheng et al., 2015, 2018,
2020; Hawkins, 2020; Lu et al., 2021), direct numerical simulations (DNSs) on
fine meshes (e.g., Stellmach et al., 2014; Aguirre Guzmán et al., 2020) or
asymptotically reduced modelling (e.g., Sprague et al., 2006; Julien et al.,
2012b; Maffei et al., 2021). A complicating factor for experiments in
particular is that at the confining sidewall a prominent mode of convection is
formed, named the wall mode, boundary zonal flow or sidewall circulation
(Zhang et al., 2020, 2021; de Wit et al., 2020; Favier & Knobloch, 2020; Lu et
al., 2021; Ecke et al., 2022; Wedi et al., 2022). This convection mode is the
first to become unstable to buoyant forcing, before bulk convection (e.g.,
Zhang & Liao, 2009), and was found to unexpectedly persist deep into the
turbulent regimes (Madonia et al., 2021). This complicates the interpretation
of results on global flow properties (like the efficiency of convective heat
transfer) and their comparison to studies without lateral confinement (i.e.
simulations on domains with periodic boundary conditions).

The difficulty of entering the geostrophic regime, particularly for
experiments and DNS, leaves it scarcely studied with many open questions
remaining, particularly considering the statistical description of this
turbulent flow from velocity measurements. Here, we contribute flow
measurements from our dedicated experimental setup TROCONVEX (Cheng et al.,
2018, 2020; Madonia et al., 2021). At a constant rotation rate, we choose
different strengths of buoyant forcing to scan the subranges of the
geostrophic regime. We quantify the strength of turbulence in terms of the
Reynolds number and compute kinetic energy spectra to identify the
distribution of energy over the spatial scales.

The remainder of this paper is organised as follows. In §2 we introduce the
dimensionless input and output parameters of RRBC and list proposed scaling
arguments for the Reynolds number. Then, the experimental methods are
explained in §3. Results on the measured Reynolds numbers are given in §4,
along with a test of the scaling arguments and a comparison to literature
data. In §5 we plot and discuss the computed kinetic energy spectra. We
present our conclusions in §6.

##  2 Theoretical background

We will first define the dimensionless input and output parameters of RRBC in
section 2.1. Then, the effects of rotation on the stability of convective flow
are treated in section 2.2. In section 2.3 we discuss scaling arguments for
the Reynolds number proposed in the literature.

###  2.1 Input and output parameters

Three dimensionless input parameters are required to describe the flow state.
The Rayleigh number $Ra$ expresses the ratio of thermal forcing to
dissipation, the Ekman number $Ek$ is the ratio of viscous to Coriolis forces,
and the Prandtl number $Pr$ describes the diffusive properties of the fluid.
These quantities are defined as

$Ra=\dfrac{g\alpha\Delta TH^{3}}{\nu\kappa}\,,\quad Ek=\dfrac{\nu}{2\Omega
H^{2}}\,\quad Pr=\dfrac{\nu}{\kappa}\,,$ (1)

where $g$ is gravitational acceleration, $\Delta T$ the applied temperature
difference between the plates and $H$ their vertical separation, $\Omega$ is
the rotation rate, and $\alpha$, $\nu$ and $\kappa$ are the thermal expansion
coefficient, kinematic viscosity and thermal diffusivity of the fluid,
respectively. A related parameter that has been used frequently to give an _a
priori_ indication of the relative importance of buoyant forcing to Coriolis
forces (see, e.g., Stevens et al., 2013a; Aurnou et al., 2020; Kunnen, 2021;
Ecke & Shishkina, 2023) is the so-called convective Rossby number

$Ro_{c}=\dfrac{U_{\mathrm{ff}}}{2\Omega H}=Ek\sqrt{\dfrac{Ra}{Pr}}\,,$ (2)

where the free-fall velocity $U_{\mathrm{ff}}=\sqrt{g\alpha\Delta TH}$
indicates the maximum flow speed that could develop in this convection system.

Two prominent output parameters are the Nusselt number $Nu$ and the Reynolds
number $Re$ which indicate the efficiency of convective heat transfer and
momentum transfer, respectively. The Nusselt number is defined as

$Nu=\dfrac{q}{q_{\mathrm{cond}}}=\dfrac{qH}{\lambda\Delta T}\,.$ (3)

The total heat flux $q$ (convection and conduction) is normalized by the
conductive flux $q_{\mathrm{cond}}=\lambda\Delta T/H$ with $\lambda$ the
thermal conductivity of the fluid. In absence of convection (a quiescent
fluid), $q=q_{\mathrm{cond}}$ and $Nu=1$; when convection sets in
$q>q_{\mathrm{cond}}$ and $Nu>1$. In our current configuration, with gravity
pointing downward, the total heat flux can be expressed as

$q=\frac{\lambda}{\kappa}\left<wT\right>-\lambda\left<\dfrac{\partial
T}{\partial z}\right>\,,$ (4)

where $z$ is the vertical coordinate pointing upward counter to gravity and
$w$ is the vertical component of velocity. Angular brackets
$\left<\ldots\right>$ denote a suitable average, either a cross-sectional
plane average at a certain height $z$ or a volume average over the flow
domain. The Reynolds number

$Re=\dfrac{UH}{\nu}$ (5)

introduces a characteristic velocity scale $U$, usually taken to be the root-
mean-squared velocity. The ultimate goal is to understand how $Nu$ and $Re$
depend on the input parameters $Ra$, $Pr$, and $Ek$. Power-law scaling
relations are expected, e.g., Julien et al. (2012a); King et al. (2013);
Maffei et al. (2021). For nonrotating convection, the phenomenological theory
developed by Grossmann & Lohse (2000, 2001, 2002, 2004) (see also Ahlers et
al., 2009; Stevens et al., 2013b) has been very successful in explaining the
$Nu(Ra,Pr)$ and $Re(Ra,Pr)$ relations.

###  2.2 Stabilisation of convection by rotation

Rotation tends to stabilise convection. For $Pr>0.68$, linear stability theory
(Chandrasekhar, 1961; Niiler & Bisshopp, 1965) predicts the critical Rayleigh
number $Ra_{C}$ for onset of convection and the characteristic horizontal size
$\ell_{C}$ of convection near onset (half the most unstable wavelength) as

$Ra_{C}=8.70Ek^{-4/3}\,,\quad\ell_{C}/H=2.41Ek^{1/3}\,.$ (6)

(The onset of convection for $Pr<0.68$ behaves differently (Chandrasekhar,
1961; Aurnou et al., 2018); it is out of scope for the current investigation.)
Due to this stabilising effect, it is insightful to also consider the degree
of supercriticality $Ra/Ra_{C}$ to get a first impression of the intensity of
convection.

###  2.3 Reynolds number scaling in rotating convection

Several scaling relations for the Reynolds number have been proposed for RRBC
flow. They are based on balancing terms in the govering Navier–Stokes equation
for an incompressible Oberbeck–Boussinesq fluid, which is (e.g.,
Chandrasekhar, 1961)

$\dfrac{\partial\boldsymbol{u}}{\partial
t}+(\boldsymbol{u}\bcdot\bnabla)\boldsymbol{u}=-\bnabla
p-2\Omega\hat{\boldsymbol{z}}\times\boldsymbol{u}+\nu\nabla^{2}\boldsymbol{u}+g\alpha
T\hat{\boldsymbol{z}}\,,\quad\bnabla\bcdot\boldsymbol{u}=0\,.$ (7)

This equation describes the evolution of velocity $\boldsymbol{u}$ as a
function of time $t$, where $p$ is the reduced pressure (e.g., Greenspan,
1968; Kundu & Cohen, 2002) and $T$ is temperature; $\hat{\boldsymbol{z}}$ is
the vertical unit vector pointing upward, counter to gravity. Centrifugal
buoyancy is excluded here (Horn & Aurnou, 2018, 2019). The equation for
vorticity $\boldsymbol{\omega}=\bnabla\times\boldsymbol{u}$ is also invoked in
these scaling arguments:

$\dfrac{\partial\boldsymbol{\omega}}{\partial
t}+(\boldsymbol{u}\bcdot\bnabla)\boldsymbol{\omega}=(\boldsymbol{\omega}\bcdot\bnabla)\boldsymbol{u}+2\Omega\dfrac{\partial\boldsymbol{u}}{\partial
z}+\nu\nabla^{2}\boldsymbol{\omega}-g\alpha\hat{\boldsymbol{z}}\times\bnabla
T\,.$ (8)

One proposed scaling relation balances the viscous, buoyancy and Coriolis
terms into the so-called visco–Archimedean–Coriolis (VAC) balance (Aubert et
al., 2001; Gillet & Jones, 2006; King et al., 2013). First, we recall the
exact global balance of energy (viscous dissipation equals buoyant production)
that can be found by averaging the energy equation $\boldsymbol{u}\bcdot\,$(7)
over the flow domain (Shraiman & Siggia, 1990; Grossmann & Lohse, 2000; Ahlers
et al., 2009):

$\nu\left<(\bnabla\boldsymbol{u})^{2}\right>_{V}=g\alpha\left<wT\right>_{V}=\dfrac{\nu^{3}}{H^{4}}\dfrac{Ra(Nu-1)}{Pr^{2}}\,,$
(9)

where $\left<\ldots\right>_{V}$ denotes a volume average. For modest
convection, geostrophy and the Taylor–Proudman theorem are considered to be
broken by viscosity acting on small horizontal scales. The balance of viscous
and Coriolis terms in the vorticity equation is scaled by recognising that
vertical derivatives $\partial/\partial z$ scale as $1/H$ and horizontal
derivatives $\partial/\partial x$ and $\partial/\partial y$ as $1/\ell$, where
$\ell\ll H$ is the horizontal scale of convection:

$2\Omega\dfrac{\partial\boldsymbol{u}}{\partial
z}\sim\nu\nabla^{2}\boldsymbol{\omega}\quad\rightarrow\quad
2\Omega\dfrac{U}{H}\sim\nu\dfrac{U}{\ell^{3}}\quad\rightarrow\quad\dfrac{\ell}{H}\sim\left(\dfrac{\nu}{2\Omega
H^{2}}\right)^{1/3}=Ek^{1/3}\,.$ (10)

The scaling of $\ell$ is in agreement with onset lengthscale $\ell_{C}$ of eq.
(6). Using $\bnabla\sim 1/\ell$ (i.e. dominated by the horizontal derivatives)
we can scale the first term in eq. (9) as $\nu U^{2}/\ell^{2}$, then, using
eq. (10), rewrite it in terms of the Reynolds number as

$\nu\dfrac{U^{2}}{\ell^{2}}\sim\dfrac{\nu^{3}}{H^{4}}\dfrac{Ra(Nu-1)}{Pr^{2}}\quad\rightarrow\quad
Re_{\mathrm{VAC}}\sim\dfrac{Ra^{1/2}Ek^{1/3}(Nu-1)^{1/2}}{Pr}\,.$ (11)

A second proposed scaling is built on a balance of
Coriolis–inertial–Archimedean (CIA) forces (Ingersoll & Pollard, 1982; Aubert
et al., 2001; Jones, 2011; Guervilly et al., 2019; Aurnou et al., 2020). For a
fully turbulent flow, effects of viscosity are negligible and Coriolis and
inertial forces are balanced in the vorticity equation:

$2\Omega\dfrac{\partial\boldsymbol{u}}{\partial
z}\sim(\boldsymbol{u}\bcdot\bnabla)\boldsymbol{\omega}\quad\rightarrow\quad
2\Omega\dfrac{U}{H}\sim\dfrac{U^{2}}{\ell^{2}}\quad\rightarrow\quad\dfrac{\ell}{H}\sim\left(\dfrac{U}{2\Omega
H}\right)^{1/2}=Ro^{1/2}\,,$ (12)

where the horizontal length scale $\ell$ now scales as the square root of the
Rossby number $Ro=U/(2\Omega H)$ based on the velocity scale $U$ (Guervilly et
al., 2019). Then, inertia and buoyancy are balanced in the energy equation:

$\boldsymbol{u}\bcdot((\boldsymbol{u}\bcdot\bnabla)\boldsymbol{u})\sim
g\alpha\boldsymbol{u}\bcdot(T\hat{\boldsymbol{z}})\quad\rightarrow\quad\dfrac{U^{3}}{\ell}\sim\dfrac{\nu^{3}}{H^{4}}\dfrac{Ra(Nu-1)}{Pr^{2}}\,,$
(13)

where (9) has been used invoking a volume average. The final scaling relation,
combining eqs. (12) and (13), can be rewritten in terms of $Re$ as

$Re_{\mathrm{CIA}}\sim\dfrac{Ra^{2/5}Ek^{1/5}(Nu-1)^{2/5}}{Pr^{4/5}}\,.$ (14)

What remains to be determined is the dependence of the Nusselt number $Nu$ on
the input parameters. For rotating convection this relation is far from
complete, with a variety of subregimes opening up in the geostrophic regime
based on recent numerical and experimental evidence (Sprague et al., 2006;
Julien et al., 2012b; Ecke & Niemela, 2014; Cheng et al., 2015, 2018, 2020; Lu
et al., 2021; Kunnen, 2021). Two theoretical scalings have been proposed. The
first (King et al., 2009, 2012) is a rotating equivalent of Malkus’s marginal-
stability argument for the thermal boundary layer (Malkus, 1954): the Rayleigh
number $Ra_{\delta}$ based on the thickness $\delta$ of the thermal boundary
layer exceeds its critical value $Ra_{\delta,C}\sim Ek_{\delta}^{-4/3}$ with
an Ekman number also based on $\delta$. With the assumption of a temperature
drop $\Delta T_{\delta}\sim\Delta T$ over a thermal boundary layer that is
purely conductive ($q\sim k\Delta T/\delta$), King et al. (2009, 2012) find

$Nu\sim\left(\dfrac{Ra}{Ra_{C}}\right)^{3}\sim Ra^{3}Ek^{4}\,.$ (15)

The second scaling relation is a rotating equivalent of Spiegel’s argument
that under vigorously turbulent conditions the heat flux $q$ should become
independent of the diffusive fluid properties $\nu$ and $\kappa$ (Spiegel,
1971). Stevenson (1979) and Julien et al. (2012a) show that, if rotation is
included, the only combination of parameters leading to diffusion-free total
heat flux $q$ is

$Nu-1\sim\dfrac{Ra^{3/2}Ek^{2}}{Pr^{1/2}}\,.$ (16)

Numerical and experimental evidence points towards the asymptotic validity of
the latter scaling (16) (Julien et al., 2012a; Stellmach et al., 2014;
Bouillaut et al., 2021), although the presence of no-slip walls (Stellmach et
al., 2014; Kunnen et al., 2016; Plumley et al., 2017; Aguirre Guzmán et al.,
2021) and the significant heat flux contribution of the wall mode near
sidewalls in confined convection (de Wit et al., 2020; Zhang et al., 2020,
2021; Favier & Knobloch, 2020; Lu et al., 2021; Ecke et al., 2022) preclude
observation of the pure scaling law.

Upon insertion of (16) into (14) we obtain in dimensionless and dimensional
form that

$Re\sim\dfrac{RaEk}{Pr}\quad\rightarrow\quad U\sim\dfrac{g\alpha\Delta
T}{2\Omega}\,,$ (17)

i.e., the velocity scale $U$ is now also diffusion-free (Aurnou et al., 2020).

##  3 Experimental methods

The experimental setup used for this study is TROCONVEX, a large-scale
rotating convection apparatus designed for the study of the geostrophic regime
of convection. The design considerations with regard to the accessible
parameter range for a given setup have been discussed in detail in Cheng et
al. (2018). Details on the setup for heat transfer measurements are given in
Cheng et al. (2020). Here we perform optical flow measurements using
stereoscopic particle image velocimetry (stereo-PIV; Prasad, 2000; Raffel et
al., 2007). The arrangement has been explained in Madonia et al. (2021); here
we repeat the most important parts.

The convection cell is an upright cylinder of height $H=2\,\mathrm{m}$ and
radius $R=0.195\,\mathrm{m}$ filled with water. Its diameter-to-height aspect
ratio is $\Gamma=2R/H=0.195$. The bottom plate is made of copper and is
electrically heated to be at a controlled temperature $T_{b}$. Likewise, the
temperature of the copper top plate, equipped with a double spiral groove for
cooling liquid circulation, is controlled to a temperature $T_{t}$ by a
combination of a chiller and a thermostated bath. The mean temperature
$T_{m}=(T_{b}+T_{t})/2$ is kept at a constant $31^{\circ}\mathrm{C}$, so that
$Pr=5.2$. The convection cell is placed on a rotating table. We apply constant
rotation $\Omega=1.93\,\mathrm{rad/s}$, corresponding to $Ek=5.00\times
10^{-8}$. The temperature difference $\Delta T=T_{b}-T_{t}$ is changed between
experiments to vary $Ra$. Additionally, one nonrotating experiment is included
for reference. The operating conditions for the various experiments are
summarised in table 1.

$\Delta T$ (${}^{\circ}\mathrm{C}$) | $Ra/10^{12}$ | $Ek\times 10^{8}$ | $Ra/Ra_{C}$ | $Re_{u}/10^{3}$ | $Re_{w}/10^{3}$
---|---|---|---|---|---
$\hphantom{0}0.50$ | $0.108$ | $5.00$ | $\hphantom{0}2.29$ | $\hphantom{0}3.10\pm 0.21$ | $\hphantom{0}2.31\pm 0.27$
$\hphantom{0}1.00$ | $0.216$ | $5.00$ | $\hphantom{0}4.57$ | $\hphantom{0}3.09\pm 0.25$ | $\hphantom{0}2.18\pm 0.20$
$\hphantom{0}2.00$ | $0.432$ | $5.00$ | $\hphantom{0}9.14$ | $\hphantom{0}3.88\pm 0.20$ | $\hphantom{0}2.63\pm 0.31$
$\hphantom{0}3.00$ | $0.648$ | $5.00$ | $13.7\hphantom{0}$ | $\hphantom{0}5.80\pm 0.80$ | $\hphantom{0}3.43\pm 0.31$
$\hphantom{0}5.00$ | $1.08\hphantom{0}$ | $5.00$ | $22.9\hphantom{0}$ | $\hphantom{0}7.93\pm 0.95$ | $\hphantom{0}5.60\pm 0.48$
$10.0\hphantom{0}$ | $2.16\hphantom{0}$ | $5.00$ | $45.7\hphantom{0}$ | $12.88\pm 0.88$ | $\hphantom{0}8.66\pm 0.46$
$20.0\hphantom{0}$ | $4.32\hphantom{0}$ | $5.00$ | $91.4\hphantom{0}$ | $17.67\pm 0.92$ | $12.90\pm 1.16$
$\hphantom{0}3.00$ | $0.648$ | $\infty$ | $O(10^{8})$ | $10.34\pm 1.61$ | $13.66\pm 2.59$
Table 1: Parameter values for the experiments. In all cases $\Gamma=0.195$ and
$Pr=5.2$. Output parameters are the Reynolds numbers $Re_{u}$ and $Re_{w}$
based on horizontal and vertical root-mean-square velocities, respectively.
For the nonrotating experiment (bottom row) the parameter $Ra/Ra_{C}$ is not
explicitly stated. It is large given that without rotation $Ra_{C}$ is only
$O(10^{3})$; quantitative comparison to the rotating cases is not meaningful.

Optical access is facilitated by a custom-made water-filled prism mounted
around the cylinder, significantly reducing refraction on the cylinder wall.
It allows for horizontal crossing of a laser light sheet with a thickness of
about $3.5\,\mathrm{mm}$ at midheight $z/H=0.5$. The laser pulses at
frequencies of $7.5$ or $15\,\mathrm{Hz}$ depending on the typical flow
speeds. The water is seeded with polyamid particles with a nominal diameter of
$5\,\mathrm{\umu m}$. Two $5\,\mathrm{MPixel}$ cameras (Jai SP-500M-CXP2)
equipped with Scheimpflug adapters are each placed at a stereoscopic angle of
approximately $45^{\circ}$ with the vertical, so that stereo-PIV can be
applied (Prasad, 2000; Raffel et al., 2007). We can measure the full velocity
vectors $\boldsymbol{u}=(u,v,w)$ in the laser light sheet plane, resulting in
a grid of vectors with a separation of $3.2\,\mathrm{mm}$ in both horizontal
directions that fits $122$ vectors in the cylinder diameter. Between $3000$
and $9000$ vector fields are evaluated per experiment for a measurement
duration ranging from $200$ to $600\,\mathrm{s}$.

Results are presented in terms of Reynolds numbers. Horizontal and vertical
velocities are treated separately. The Reynolds number $Re_{u}$ is based on
the characteristic horizontal velocity
$U_{h}=\sqrt{\left<u^{2}\right>+\left<v^{2}\right>}$. Here
$\left<\ldots\right>$ either denotes averaging over time and over circular
shells to obtain radial profiles, or averaging over time and over the full
cross-sectional area excluding the wall mode near the sidewall (to be more
precisely defined later on) to get to a single representative value per
experiment. In this azimuthal averaging, we divide the section in 65
concentric circular shells. Each shell represents a ring of thickness
$3\,\mathrm{mm}$ comparable to the resolution of our PIV vector field spacing
($3.2\,\mathrm{mm}$). Similarly, $Re_{w}$ is based on the characteristic
velocity scale $W=\sqrt{\left<w^{2}\right>}$. In all cases
$\left<u\right>\approx\left<v\right>\approx\left<w\right>\approx 0$. These
Reynolds numbers can also be interpreted as the root mean square (rms)
velocities normalised with the viscous velocity scale $U_{\nu}=\nu/H$.
Likewise, we consider the rms value of the vertical component of vorticity,
$\omega_{z}^{\mathrm{rms}}$.

An important consideration that is encountered in rotating convection
experiments is the effect of centrifugal buoyancy (Horn & Aurnou, 2018, 2019).
We use the Froude number $Fr=\Omega^{2}R/g$ to quantify the ratio of maximal
centrifugal to gravitational acceleration. Since centrifugal acceleration is
negligibly small in most geophysical and astrophysical applications, its
effects (as studied in detail by Horn & Aurnou, 2018, 2019) should preferably
be kept to a minimum in experiments, i.e. $Fr\ll 1$. Here, $Fr=0.074$ for all
rotating experiments. Such $Fr$ did not lead to significant up/down asymmetry
in our sidewall temperature measurements (Cheng et al., 2020); we expect that
centrifugal buoyancy is negligibly small in these experiments, too.

To give a first impression of the measurement results, we plot two
instantaneous velocity field snapshots at $Ra=6.48\times 10^{11}$, with and
without rotation, in figure 1. Velocities are normalised with the viscous
velocity scale $\nu/H$, which is $3.87\times 10^{-7}\,\mathrm{m/s}$ at this
mean temperature. As such, the magnitude of the normalised velocity components
in these plots can be interpreted as displaying the local instantaneous
Reynolds number. It is clear that rotation, present in panel (a) but absent in
panel (b), has a large effect on the overall flow structuring. Under rotation
the flow tends to organise into structures considerably smaller than the
diameter of the cylinder, while without rotation the global organisation is as
large as the cross-section. We have considered the characteristic size of the
flow features before in Madonia et al. (2021), where we could see that the
correlation length for vertical velocity increases as $Ra$ grows. In the
rotating experiments we can also observe intense vertical motion near the
sidewall due to the convective wall mode (de Wit et al., 2020; Zhang et al.,
2020, 2021; Favier & Knobloch, 2020; Lu et al., 2021; Ecke et al., 2022). For
roughly half of the circumference we find upward flow close to the sidewall;
the other half downward. In the nonrotating experiment (figure 1(b)) the
large-scale circulation is observed (e.g., Ahlers et al., 2009): a vertical
convection roll filling the domain with one half upward and one half downward
flow.

Figure 1: Instantaneous velocity snapshots at $Ra=6.48\times 10^{11}$: (a)
with rotation, $Ek=5.00\times 10^{-8}$, (b) no rotation, $Ek=\infty$. The
arrows display the horizontal velocity components; for clarity only one ninth
of the total number of vectors is displayed. The background colour indicates
the vertical velocity component. Velocities are normalised with the viscous
velocity $\nu/H=3.87\times 10^{-7}\,\mathrm{m/s}$.

##  4 Reynolds number results

###  4.1 Radial profiles

Figure 2: (a) Radial dependence of $Re_{u}$ (dashed lines) and $Re_{w}$ (solid
lines) for different $Ra$. Black crosses indicate the beginning of the
sidewall boundary layer following 18. (b) Radial dependence of
$\omega_{z}^{\mathrm{rms}}$ for different $Ra$, normalised using the viscous
time scale $\tau_{\nu}=H^{2}/\nu$. The legend entry ‘NR’ refers to the
nonrotating experiment. All the quantities displayed in this figure are only
shown from half the cylinder radius onward for clarity.

We first consider radial profiles of the Reynolds numbers $Re_{u}$ and
$Re_{w}$, plotted in figure 2(a). Both quantities show an overall trend of
larger values for larger $Ra$, as expected, with the exception of the two
lowest $Ra$ cases that are almost identical. The distinction between an inner
and a near-wall outer portion of the domain is clear from both quantities:
both profiles suggest a generally constant value that is largely independent
of the radial position for the bulk part, away from the lateral sidewall, and
a change of behavior close to this sidewall. There we see an increase in
$Re_{w}$, a clear indication of the presence of vigorous convection near the
sidewall in the wall mode. The black crosses in figure 2(a) indicate the
approximate thickness of the sidewall boundary layer following the empirical
relation

$\delta_{w_{\mathrm{rms}}}/R=aRa^{0.15\pm 0.02}\,,$ (18)

based on capturing the radial extent of the near-wall $w_{\mathrm{rms}}$ peak;
$a=(3\pm 1)\times 10^{-3}$ is reported by de Wit et al. (2020). Here we use
$a=2\times 10^{-3}$ as the prefactor and $0.15$ as the exponent for $Ra$. This
definition agrees well with our data as it indicates the extent of the
sidewall layer, in both $Re_{w}$ and $Re_{u}$. For the former it is at the
base of the near-wall peak, for the latter we see the beginning of the decay
to zero.

The non-rotating case (black curves in figure 2(a)) shows three main points of
interest. First, $Re_{w}$ displays an immediate decay to zero close to the
sidewall without any previous increase, contrary to the rotating cases.
Second, without rotation we measure significantly higher values of both
$Re_{u}$ and $Re_{w}$ relative to the rotating case at the same $Ra$ (compare
black and purple curves in figure 2(a)), a clear indication that the strong
influence of rotation makes the flow less turbulent, suppressing both vertical
and horizontal rms velocities. Third, the rms vertical velocity is now
considerably more pronounced than the horizontal component, unlike the
rotating cases. We will discuss flow anisotropy later in this paper.

In figure 2(b) we plot $\omega_{z}^{\mathrm{rms}}$ normalised using the
viscous time scale $\tau_{\nu}=H^{2}/\nu$. The same trends as for $Re_{w}$ are
reflected here, with the two cases with lowest $Ra$ being almost identical and
profiles that increase for higher $Ra$. Also here we see higher values for the
non-rotating case compared to its corresponding rotating counterpart, even
though the difference is less than the one we see for vertical velocities. The
non-rotating case also shows a localised peak close to the sidewall, while the
rotating cases have a wider radial region where the vorticity increases before
dropping down at the wall.

All the quantities displayed in figure 2 are shown from half the cylinder
radius onward for clarity. The inner part keeps showing a constant behavior
down to approximately $\frac{1}{10}R$, a circle of around $20\,\mathrm{mm}$
around the axis of the cylinder where the scarcity of velocity vectors
prevents meaningful radial binning and azimuthal averaging.

The quantities $Re_{u}$, $Re_{w}$ and $\omega_{z}^{\mathrm{rms}}$ are
approximately constant in the inner part of the cylinder. From the profiles in
figure 2 we can extract for every $Ra$ a mean value of that quantity in the
bulk. This bulk average is obtained by excluding the wall zone as defined by
equation (18) as well as excluding a circle of radius $20\,\mathrm{mm}$ from
the cylinder axis, an area where the azimuthal averaging does not give
trustworthy data, as argued above. In figure 3(a) we plot these averaged data
as a function of $Ra$, with error bars that represent the standard deviations
of these means. We also give the corresponding flow states as inferred from
our earlier heat transfer measurements (Cheng et al., 2020).

Figure 3: (a) Dependence on $Ra$ of Reynolds numbers based on horizontal
($Re_{u}$) and vertical ($Re_{w}$) velocity components (red, left ordinate)
and vertical vorticities ($\omega_{z}^{\mathrm{rms}}$; blue, right ordinate).
Vorticity is normalised with the viscous timescale $\tau_{\nu}=H^{2}/\nu$. The
solid lines represent power-law fits $Re_{u}\sim Ra^{0.65\pm 0.07}$,
$Re_{w}\sim Ra^{0.70\pm 0.06}$ and $\omega_{z}^{\mathrm{rms}}\sim Ra^{0.63\pm
0.05}$. (b) Kinetic energy anisotropy $A=W^{2}/(U_{h}^{2}+W^{2})$ versus $Ra$.
For isotropy $A=\tfrac{1}{3}$ (red dashed line). In both panels the dotted and
the dash-dotted lines represent the transitional $Ra$ between CTCs and plumes
and between GT and RIT, respectively, while open symbols represent the values
for the nonrotating case.

As mentioned before, the two lower $Ra$ cases, both in the CTC regime, show
very similar values. From the plumes regime onward, all the quantities display
a steeper trend, that is reasonably constant between the plumes/GT and RIT
states: $Re_{u}\sim Ra^{0.65\pm 0.07}$, $Re_{w}\sim Ra^{0.70\pm 0.06}$ and
$\omega_{z}^{\mathrm{rms}}\sim Ra^{0.63\pm 0.05}$. These exponents are notably
larger than the nonrotating trend $Re\sim Ra^{0.44}$ that has been measured in
various experiments and employing different methods, as summarised by Ahlers
et al. (2009). They are even larger than the “ultimate” diffusion-free scaling
$Re\sim Ra^{1/2}$ for nonrotating convection (Kraichnan, 1962; Spiegel, 1971;
Grossmann & Lohse, 2002). Another clear difference between the current
nonrotating and rotating experiments is that without rotation the values are
higher than with rotation at the same $Ra$.

The vorticity scaling $\omega_{z}^{\mathrm{rms}}\sim Ra^{0.63}$ nicely follows
that of horizontal velocity. From this information we can infer that the
characteristic horizontal length scale $\ell$ does not change much with $Ra$,
employing the estimate $\omega_{z}\sim U_{h}/\ell$. This $Ra$-independent
characteristic scale for rapidly rotating convection is one of the starting
points of the asymptotically reduced equations (e.g. Sprague et al., 2006;
Julien et al., 2012b) that describe the flow in the limit $Ek\rightarrow 0$
and that have provided a lot of insights into geostrophic convection. We have
measured lengthscales before in Madonia et al. (2021), verifying the
$Ra$-independence of $\ell$.

To study the degree of anisotropy in the bulk flow, we plot in figure 3(b) the
kinetic energy anisotropy for each case, defined as

$A=\dfrac{W^{2}}{U_{h}^{2}+W^{2}}\,.$

It describes the fraction of total kinetic energy found in vertical motions.
In the isotropic case, where the energy is equally distributed among the three
components, this value would be $\frac{1}{3}$. As we see from figure 3(b), the
non-rotating case is far away from that value, while the rotating cases
display values very close to the isotropic one, with the possible exception of
the case with $Ra=6.48\times 10^{11}$, possibly a point of transition between
the plumes and geostrophic-turbulence (GT) states of the geostrophic regime,
for which no transition relation is currently available. Here, rotation thus
suppresses the large anisotropy of nonrotating convection, ending up in near-
isotropy.

###  4.2 Test of force balance scaling relations

We can use the measured characteristic velocities to test compliance (in terms
of $Ra$-scaling) with the scaling relations (11) and (14), respectively based
on the VAC and CIA force balances. We graphically test the compliance in
figure 4 by plotting $Re/Re_{\mathrm{VAC}}$ and $Re/Re_{\mathrm{CIA}}$. Input
on the $Nu(Ra)$ scaling is required. Here, we use our prior heat transfer
results obtained in the same experimental setup by Cheng et al. (2020), viz.
$Nu\sim Ra^{0.64}$ for the plumes/GT range and $Nu\sim Ra^{0.52}$ for the RIT
range. It must be emphasised that these measured relations may be affected by
the heat transfer contribution of the sidewall circulation (de Wit et al.,
2020; Zhang et al., 2020; Lu et al., 2021). We also include the ultimate
scaling $Nu\sim Ra^{3/2}$ as per equation (16). The VAC scaling test is
included for this diffusion-free relation, too, despite the incompatibility of
combining a diffusion-free $Nu$ scaling with a force balance that explicitly
includes the viscous force. The $Re$ scaling relations (11) and (14) do not
have numerical prefactors. To make these plots, we have chosen the prefactors
such that the points are at the same level on average for ease of comparison.

Figure 4: Test of $Re(Ra)$ scaling with VAC (eq. (11)) and CIA (eq. (14))
force balance arguments, for (a,b) $Re_{u}$ and (c,d) $Re_{w}$. Three
different $Nu(Ra)$ relations are invoked: $Nu\sim Ra^{0.52}$ (green; for RIT)
and $Nu\sim Ra^{0.64}$ (red; for plumes/GT) based on our heat transfer
measurements in the same setup (Cheng et al., 2020); and $Nu\sim Ra^{3/2}$
(blue) following eq. (16).

For both $Re_{u}$ and $Re_{w}$, it is observed that either VAC or CIA scaling
does a good job of describing the results, provided our earlier experimental
$Nu(Ra)$ relations are employed. We see that both relations $Nu\sim Ra^{0.52}$
and $Nu\sim Ra^{0.64}$ lead to comparable results and are equally valid in the
plumes/GT and RIT ranges. Based on these plots and on the current data, we
cannot decide which scaling (VAC or CIA) is most appropriate. This is in line
with the analysis of Hawkins (2020), who similarly could not reach a
conclusion based on measurements using laser Doppler velocimetry. We can
insert these numbers into the scaling relations (11) and (14) for a
quantitative comparison. Using $Nu\sim Ra^{0.64}$ for the plumes/GT range, we
arrive at the prediction $Re\sim Ra^{0.82}$ from VAC scaling and $Re\sim
Ra^{0.66}$ from the CIA balance. When we use $Nu\sim Ra^{0.52}$ as obtained
for the RIT range, we find $Re\sim Ra^{0.76}$ for VAC and $Re\sim Ra^{0.61}$
for CIA. Our exponents for the velocity scalings from figure 3 are in this
range, perhaps a bit closer to the CIA trend than to the VAC scaling, but
certainly not giving a conclusive answer either. We are probably in a state
where both inertial and viscous forces play a role. The flow is turbulent
enough that inertial forces are relevant, but not yet turbulent enough to
reach the diffusion-free scaling $Re\sim Ra$ of eq. (17).

To further interpret our data and compare to previously published results, we
collect several reported datasets for $Re_{w}$ in figure 5. We plot
$\widetilde{Re}_{w}=Re_{w}Ek^{1/3}$ as a function of
$(\widetilde{Ra}-\widetilde{Ra}_{C})/Pr=(Ra-Ra_{C})Ek^{4/3}/Pr$, both inspired
and necessitated by the results of Maffei et al. (2021) from asymptotic model
simulations. In their asymptotic equation $Ek\rightarrow 0$ so $Ek$ itself
remains undefined. Furthermore, using this plot convention they obtained a
good collapse of their data spanning a range of $Pr$ and $\widetilde{Ra}$, see
the plus symbols in the plot. Their simulation domain is a Cartesian box with
periodic boundary conditions in the horizontal directions and stress-free
walls on bottom and top. The other datasets are from DNS and experiments,
where the corresponding Ekman number is colour coded. The current results are
included with diamonds. The experimental results by Hawkins (2020) cover a
significant range of $Ra\simeq 10^{8}-2\times 10^{11}$ and $Ek\simeq
10^{-7}-3\times 10^{-5}$; they are the reported RRBC Reynolds number data
closest to the current study in terms of $Ra$ and $Ek$ and are included with
dots. Numerical and experimental results corresponding to a smaller
cylindrical convection cell at $Ra\approx 10^{9}$ and $Ek\simeq 3\times
10^{-6}-10^{-3}$ taken from Kunnen et al. (2010) and Rajaei et al. (2018) are
included with up and down triangles. Finally, the DNS study by Aguirre Guzmán
et al. (2020) on a horizontally periodic domain with no-slip walls on bottom
and top at $Ra=5\times 10^{9}-10^{12}$ and $Ek\approx 10^{-7}$ is included
with open circles. From this figure, we can make several observations.

Figure 5: Comparison of current results for $Re_{w}$ with published results.
Plusses: data from Maffei et al. (2021), colour coded by $Pr$, results from
asymptotically reduced model simulations on a horizontally periodic domain.
Other symbols colour coded by $Ek$ (see colour bar). Diamonds: current
results. Dots: experiments of Hawkins (2020). Up triangles: DNS in a cylinder
(Kunnen et al., 2010). Down triangles: experiments of Rajaei et al. (2018).
Circles: DNS in a horizontally periodic domain (Aguirre Guzmán et al., 2020).

First, the plateauing at low $Ra$ that we observed in figure 3(a) is also
observed in the data of Hawkins (2020). On the contrary, the simulation
results on horizontally periodic domains (Aguirre Guzmán et al., 2020; Maffei
et al., 2021) do not show such a plateau. The principal distinguishing
property is the presence or absence of a sidewall. We know that the sidewall
circulation generates jet-like intrusions from the sidewall region into the
bulk (Madonia et al., 2021). It is thus plausible that the observed plateauing
of $Re_{w}$ is caused by the dominance of the jets emerging from the sidewall
circulation over the columnar convection in the central region. Note that the
difference between the DNS of Aguirre Guzmán et al. (2020) and the asymptotic
model of Maffei et al. (2021) at low $\widetilde{Ra}=RaEk^{4/3}$ is related to
the applied boundary conditions: in this range, paradoxically, no-slip
boundaries lead to higher $Nu$ (and thus also higher $Re_{w}$) than stress-
free boundaries at otherwise identical parameter values (Schmitz & Tilgner,
2010; Stellmach et al., 2014; Kunnen et al., 2016; Plumley et al., 2017). This
effect has been ascribed to the prevalence of Ekman pumping as a source of
vertical transport, present for no-slip plates but absent for stress-free.

Second, the Reynolds number scaling depends sensitively on the magnitude of
$Ra$ and $Ek$. The smaller-scale cylinder investigations of Kunnen et al.
(2010) and Rajaei et al. (2018) at moderate $Ra\approx 10^{9}$ and $Ek$ down
to $3\times 10^{-6}$ display a significantly shallower slope than the larger
cylinders (larger $Ra$ and smaller $Ek$) of Hawkins (2020) and the current
study. There is definitely a different scaling upon entering the geostrophic
regime of rotating convection. In fact, these four studies together display a
gradual transition as a function of both $Ra$ and $Ek$. A steeper scaling can
be seen as $Ek$ is reduced, while concomitantly increasing $Ra$ to retain
turbulent convection, save for the added complications due to the sidewall
circulation.

Third, the smallest-$Ek$ datapoints (orange and red) display a trend towards
convergence to the asymptotic results for $(Ra-Ra_{C})Ek^{4/3}/Pr\gtrsim 50$
and $Re_{w}Ek^{1/3}\gtrsim 20$. Notably, results of completely different
origin (experiments, DNS and asympotic models; horizontally periodic domains
and confined cylinders) display convergence towards a common scaling
behaviour. This can be taken as a strong indicator that the “ultimate”
diffusion-free scaling is approached. A linear scaling in this diagram
directly corresponds to the relation expressed in eq. (17) where the velocity
scale becomes independent of both diffusive parameters and domain-specific
parameters like its height $H$. The convergence is not yet fully achieved;
extra datapoints at even smaller $Ek$ are required to corroborate this, but
tough to realise experimentally or numerically.

##  5 Energy spectra

To compute the energy spectra we employ two-dimensional fast Fourier
transforms of the velocity field, after padding the vector fields with zeros
up to 512 grid points per dimension to avoid artifacts due to implied
periodicity in the Fourier transform (Randall, 2008). The two-dimensional data
is averaged over circular shells to obtain one representative curve
independent of azimuthal angle. Since adjacent velocity vectors are $\Delta
x=3.2\,\mathrm{mm}$ apart, the smallest wavenumber after zero padding is
$k_{\mathcal{L}}=2\pi/\mathcal{L}=2\pi/(512\Delta x)\approx
3.83\,\mathrm{m^{-1}}$, and the maximum relevant $k$ value (Nyquist
wavenumber) is $k_{2\Delta x}=2\pi/(2\Delta x)\approx 982\,\mathrm{m^{-1}}$.
All the $k$ values are integer multiples of $k_{\mathcal{L}}$. Nevertheless,
the smallest physically relevant $k$ is determined by the diameter $D$ of the
cylinder: that would be given by $k_{D}=2\pi/D\approx 16.1\,\mathrm{m^{-1}}$,
but since all the wavenumbers are integer multiples of $k_{\mathcal{L}}$, our
first relevant value is $k_{\mathrm{min}}\approx 19.2\,\mathrm{m^{-1}}$, in
dimensionless form $k_{\mathrm{min}}H\approx 38.4$.

The maximum relevant $k$ requires more consideration. Previous literature
(Foucaut et al., 2004; Savelsberg, 2006) has shown that the window averaging
applied in PIV measurements acts as a low-pass filter in Fourier space with a
cut-off wavenumber that depends on the size of the measurement window $X$ (in
our case $X=2\Delta x=6.4\,\mathrm{mm}$). The PIV filtering amounts to a
multiplication in Fourier space of the real signal with a squared cardinal
sine function

$\mathrm{sinc}^{2}(kX/2)=\left(\frac{\sin{kX/2}}{kX/2}\right)^{2}\,.$ (19)

We can then correct for this filtering, with corrected energy spectra becoming

$E_{\mathrm{corr}}=\frac{E_{\mathrm{meas}}}{\mathrm{sinc}^{2}(kX/2)}\,,$ (20)

where $E_{\mathrm{meas}}$ is the uncorrected spectrum determined directly from
the PIV data and $E_{\mathrm{corr}}$ is the corrected spectrum. However, given
that $\mathrm{sinc}^{2}(kX/2)\rightarrow 0$ as $kX/2\rightarrow\pi$, direct
application of eq. (20) may cause practical difficulties: measurement noise
will be severely amplified when approaching the zero of the denominator. To
deal with this, we employ the rule of thumb proposed by Foucaut et al. (2004)
to cut off the spectra at a maximum $k_{\mathrm{max}}=2.8/X$. At this $k$
value the filter function $\mathrm{sinc}^{2}(k_{max}X/2)\approx 0.5$, i.e. the
spectrum is cut off at the wavenumber $k_{\mathrm{max}}$ where PIV filtering
reduces the measured spectral amplitude by $50\%$. In our case
$k_{\mathrm{max}}H=874$.

Figure 6: Kinetic energy spectra $E(k)$ plotted as a function of normalised
wavenumber $kH$, including total kinetic enery ($E_{\mathrm{tot}}$) as well as
the contributions from horizontal ($E_{\mathrm{hor}}$) and vertical
($E_{\mathrm{ver}}$) velocity components. Reference power-law slopes
$k^{-5/3}$ and $k^{-3}$ are also included.

In figure 6 we show the energy spectra of our seven RRBC cases as well as the
non-rotating case. We plot the total energy spectrum ($E_{\mathrm{tot}}$)
together with the respective contributions from the horizontal
($E_{\mathrm{hor}}$) and vertical ($E_{\mathrm{ver}}$) velocity components.
Wavenumbers are normalised using the cell height $H$.

The rotating cases show common features. In all of them, $E_{\mathrm{tot}}$
exhibits a peak at $kH\approx 70$, which corresponds to the physical
wavelength $L\approx 0.18\,\mathrm{m}\approx D/2$. This peak is related to the
ordering into a quadrupolar vortex state that we reported before (Madonia et
al., 2021), which in hindsight was also found in the mean flow fields of de
Wit et al. (2020). Ecke et al. (2022) also show this flow organisation in
their figure 2(c). Jets pointing radially inward are observed at the positions
where up- and downward flowing parts of the sidewall circulation meet,
resulting in a four-quadrant organisation with two cyclonic and two
anticyclonic vortices. The formation of these jets is a nonlinear feature of
the sidewall circulation absent from its linear description (Herrmann & Busse,
1993; Liao et al., 2006; Zhang & Liao, 2009). The dependence on the
geometrical aspect ratio $\Gamma$ is an open question; we do anticipate
changes for wider cylinders where more than one azimuthal wavelength can fit
into the circumference. Here, with one azimuthal wavelength, we see an
organisation into a quadrupolar vortex where each pole has a characteristic
size $L\approx D/2$. These structures are most energetic, but at the same time
they are quasi-two-dimensional: there is no signature of them in the spectra
$E_{\mathrm{ver}}$ based on vertical velocity.

The shape of the remaining part of each spectrum is strongly dependent on the
vertical energy contribution. As $Ra$ increases, the peak in
$E_{\mathrm{ver}}$ goes towards smaller $k$ (larger length scales), a feature
that we have observed before (Madonia et al., 2021) from calculations of the
characteristic horizontal correlation length scale based on vertical velocity.
We interpret this as the scale at which buoyancy is most prominently adding
energy into the turbulent motion. From the maximum of $E_{\mathrm{ver}}$
towards larger $k$ values, we see the gradual development of a spectral
scaling range close to $E(k)\sim k^{-5/3}$, the predicted scaling for the
inertial range of three-dimensional turbulence (Frisch, 1995; Pope, 2000).
This scaling range is most prominent in the $E_{\mathrm{tot}}$ curves and at
larger $Ra$. The sum of horizontal and vertical contributions is clearly
reflected in the total spectra $E_{\mathrm{tot}}$, with the peak due to the
quadrupolar vortex dominating at smaller $k$ and a broad shoulder due to the
peak contribution of $E_{\mathrm{ver}}$ that shifts as a function of $Ra$.

The horizontal spectra, in fact, also show a different scaling trend, from the
approximate location of the peak of $E_{\mathrm{ver}}$ to smaller wavenumbers
ending on the maximum of $E_{\mathrm{hor}}$, that resembles $E(k)\sim k^{-3}$
scaling. This $k^{-3}$ scaling could be a sign of a (nonlocal) inverse cascade
(Smith & Waleffe, 1999; Lindborg, 2007; Julien et al., 2012b; Aguirre Guzmán
et al., 2020), although this scaling range is short and certainly cannot be
considered as a proof. We are investigating the energy transfer among scales
in more detail.

The nonrotating RBC case, instead, shows a very different situation. The peak
of the overall spectrum is at $k_{\mathrm{min}}$, and it is dominantly due to
the contribution of $E_{\mathrm{ver}}$. It is a sign of the presence of the
large-scale circulation of nonrotating convection (e.g., Ahlers et al., 2009):
a domain-filling vertical convection roll with one half of the domain
displaying upward flow and the other half downward. We have observed it in the
current measurements (Madonia et al., 2021; Madonia, 2022). This structure
corresponds to a wavelength of approximately the diameter of the cylinder,
that in wavenumber space is translated to $k_{\mathrm{min}}$. Here we also
observe a rather extensive scaling range nicely matching the theoretical
prediction $k^{-5/3}$.

##  6 Conclusion

We present flow statistics that result from the stereo-PIV measurements of
rotating Rayleigh–Bénard convection (RRBC) in the geostrophic regime. We
analyse radial profiles of the root-mean-square (rms) values of horizontal and
vertical velocity as well as vertical vorticity. Near the sidewall, the radial
extent of the sidewall circulation can be clearly distinguished. We see that
for both velocities and vorticities higher $Ra$ corresponds to larger values
of these quantities, and at the same $Ra$ the non-rotating case exhibits
larger magnitudes than the corresponding rotating one.

Within the bulk region, for the flow states from plumes to rotation-influenced
turbulence (RIT), the Reynolds number scalings follow a trend that lies in the
range of both CIA and VAC scaling predictions, perhaps being numerically
closer to the former, but otherwise no decisive distinction can be made. The
vorticity scaling follows the same trend as the horizontal velocity,
indicating that the characteristic horizontal scale remains constant (e.g.,
Sprague et al., 2006; Julien et al., 2012b; Madonia et al., 2021). We also
find that energy is distributed nearly isotropically, in contrast to
nonrotating RBC where the vertical velocity component is strongly preferred.

The Reynolds number results compared to previously published results reveal
that there is an overall trend towards “ultimate” diffusion-free scaling
(Aurnou et al., 2020) as the Ekman number is reduced, but we do not reach it
yet. However, there are indications of convergence in scaling for results
coming from different methods (experiments, direct numerical simulations and
asymptotically reduced models) and different geometries (confined cylinders
and horizontally periodic computational domains). At lower $Ra$ there are
complications due to the use of a confined geometry, as it promotes the
presence of a wall mode that affects the bulk flow statistics.

Spectra of kinetic energy provide further evidence for the formation of a
prominent quadrupolar vortex structure (Madonia et al., 2021). This feature is
remarkably two-dimensional, as it is reflected only in spectra of horizontal
velocity. In the spectra of vertical velocity we observe a broad peak, that we
associate with the dominant length scale for energy injection. With increasing
$Ra$, the peak shifts towards smaller wavenumbers (larger length scales;
Madonia et al., 2021). For larger wavenumbers we observe the development of a
scaling range with a classical $-5/3$ slope for three-dimensional turbulence
(Frisch, 1995; Pope, 2000). This inertial-range scaling is most prominent for
the nonrotating case.

These results provide valuable input on the turbulence scaling of flows in the
geostrophic regime, a property that is scarcely investigated up to now. There
is clear perspective in the observed convergence towards diffusion-free
scaling. At the same time, to reach true convergence even more extreme
parameter values than those used here are likely required, which is a major
challenge for both experiments and direct numerical simulations. Additionally,
the influence of the strong convection mode near the sidewall, the sidewall
circulation, should be further explored to ensure proper interpretation of
experimental and numerical results from confined domains.

[Funding]M.M., A.J.A.G. and R.P.J.K. received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant Agreement No. 678634). We are grateful for the
support of the Netherlands Organisation for Scientific Research (NWO) for the
use of supercomputer facilities (Cartesius and Snellius) under Grants No.
2019.005, No. 2020.009 and No. 2021.009.

[Declaration of interests]The authors report no conflict of interest.

[Data availability statement]The data that support the findings of this study
are available from the corresponding author, R.P.J.K., upon reasonable
request.

[Author ORCIDs]Matteo Madonia, https://orcid.org/0000-0002-8510-8103; Andrés
J. Aguirre Guzmán, https://orcid.org/0000-0002-4942-5216; Herman J.H. Clercx,
https://orcid.org/0000-0001-8769-0435; Rudie P.J. Kunnen,
https://orcid.org/0000-0002-1219-694X.

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