#  Internally heated convection with rotation: bounds and scaling laws for
heat transport

Ali Arslan<EMAIL_ADDRESS>Institute of Geophysics, ETH Zürich,
Zürich, CH-8092, Switzerland

###### Abstract

This work investigates heat transport in rotating internally heated
convection, for a horizontally periodic fluid between parallel plates under
no-slip and isothermal boundary conditions. The main results are the proof of
bounds on the mean temperature, $\overline{\left\langle T\right\rangle}$, and
the heat flux out of the bottom boundary, $\mathcal{F}_{B}$ at infinite
Prandtl numbers where the Prandtl number is the nondimensional ratio of
viscous to thermal diffusion. The lower bounds are functions of a Rayleigh
number quantifying the ratio of internal heating to diffusion and the Ekman
number, $E$, which quantifies the ratio of viscous diffusion to rotation. We
utilise two different estimates on the vertical velocity, $w$, one pointwise
in the domain (Yan, 2004, _J. Math. Phys._ , vol.45(7), pp.2718-2743) and the
other an integral estimate over the domain (Constantin et al., 1999, _Phys. D:
Non. Phen._ , vol. 125, pp. 275-284), resulting in bounds valid for different
regions of buoyancy-to-rotation dominated convection. Furthermore, we
demonstrate that similar to rotating Rayleigh-Bénard convection, for small
$E$, the critical Rayleigh number for the onset of convection asymptotically
scales as $E^{-4/3}$.This result is combined with heuristic arguments for
internally heated and rotating convection to arrive at scaling laws for
$\overline{\left\langle T\right\rangle}$ and $\mathcal{F}_{B}$ valid for
arbitrary Prandtl numbers.

###### keywords:

turbulent convection, variational methods

##  1 Introduction

Heat transport by turbulent convection remains a pertinent area of research in
both astrophysical and geophysical fluid dynamics. While boundary-forced
thermal convection has been extensively studied, convection driven by internal
heating has been relatively overlooked (Doering, 2020). Nevertheless,
internally heated convection (IHC) plays a significant role within planetary
bodies, such as in the Earth’s mantle and core, where the radioactive decay of
isotopes and secular cooling drive fluid motion (Schubert et al., 2001;
Schubert, 2015). Similarly, for stars, convective zones are driven by
radiation from nuclear fusion (Schumacher & Sreenivasan, 2020) and supernovae
are modelled as fluids heated internally by neutrinos (Herant et al., 1994;
Radice et al., 2016). Moreover, stars and planets are rotating bodies and the
Coriolis force significantly affects the flow dynamics (Greenspan, 1968).

Studying rotating turbulent convection is challenging because experiments and
numerical simulations cannot reach parameter values of interest (Glatzmaier,
2013). For example, in planetary mantles, the Prandtl number, $Pr$, the non-
dimensional number quantifying the ratio of the viscous and thermal
diffusivity, reaches values of $10^{23}$, while the Rayleigh number, $R$,
quantifying the ratio of thermal forcing to diffusion is at least $10^{6}$
(Mulyukova & Bercovici, 2020). In planetary cores, the Rayleigh number could
be as high as $10^{26}$ (Schubert, 2015). Furthermore, the Ekman number, $E$,
representing the viscous to rotational forces, is estimated to be $10^{-15}$
in the Earth’s core (Jones & Schubert, 2015).

$g$$x$$y$$z$$z=0$$z=1$$T=0$$T=0$${\color[rgb]{0.636364,0.000000,0.000000}\definecolor[named]{pgfstrokecolor}{rgb}{0.636364,0.000000,0.000000}\mathcal{F}_{B}}$${\color[rgb]{0.636364,0.000000,0.000000}\definecolor[named]{pgfstrokecolor}{rgb}{0.636364,0.000000,0.000000}\overline{\left\langle
T\right\rangle}}$${\color[rgb]{0.636364,0.000000,0.000000}\definecolor[named]{pgfstrokecolor}{rgb}{0.636364,0.000000,0.000000}\mathcal{F}_{T}}$$H=1$
Figure 1:  A non-dimensional schematic diagram for rotating uniform internally
heated convection. The upper and lower plates are at the same temperature, and
the domain is periodic in the $x$ and $y$ directions and rotates about the $z$
axis. $\mathcal{F}_{B}$ and $\mathcal{F}_{T}$ are the mean heat fluxes out the
bottom and top plates, $\overline{\left\langle T\right\rangle}$ the mean
temperature, and $g$ is the acceleration due to gravity.

An alternative route for inquiry is a mathematically rigorous study of the
equations describing rotating convection. Of interest is the regime where the
solutions of the governing equations are turbulent, and a key question is on
the long-time behaviour of the mean quantities of the flow as a function of
the control parameters ($Pr,R,E$). In this study, we employ the background
field method (Doering & Constantin, 1992, 1994; Constantin & Doering, 1995;
Doering & Constantin, 1996) to study the mean heat transport in IHC subject to
rotation between parallel plates with isothermal and no-slip boundary
conditions (fig. 1). Unlike turbulent convection driven by boundary heating,
i.e. Rayleigh-Bénard convection (RBC), proving results for IHC is more
difficult, with no known rigorous results for IHC subject to rotation.

The influence of rotation alters turbulent convection and introduces new flow
regimes and physics (see Ecke & Shishkina (2023) for a recent review). The
flow features of rotating convection in a plane layer are well-established for
boundary heating (Chandrasekhar, 1961; Veronis, 1959; Rossby, 1969; Boubnov &
Golitsyn, 2012; Julien et al., 1996; Knobloch, 1998; Vorobieff & Ecke, 2002;
Stevens et al., 2013), and some insight exists for non-uniform IHC (Barker et
al., 2014; Currie et al., 2020; Hadjerci et al., 2024). However, no study has
explored the flow in rotating uniform IHC. We know that rotation inhibits the
onset of convective motion and stabilises the fluid, causing a bias in motion
parallel to the axis of rotation and creating an Ekman boundary layer that
enhances the mean vertical heat transport by Ekman pumping (Greenspan, 1968).
With sufficient thermal forcing, the $E-R$ parameter space contains two
extreme flow states: if $R$ is sufficiently larger than $E^{-1}$, then
buoyancy dominates and rotation plays little effect on the dynamics, whereas
if $E^{-1}$ is large relative to $R$, and the vertical velocity is nonzero,
geostrophic turbulence occurs (Julien et al., 1996; Sprague et al., 2006). A
wide range of flow features occur in rotating convection including, cellular
flows, Taylor columns, large-scale vortices and plume-dominated convection
(Grooms et al., 2010; Julien et al., 2012; Stellmach et al., 2014; Guzmán et
al., 2020; Aurnou et al., 2020; Song et al., 2024).

In addition to numerous experimental and numerical studies on rotating RBC,
there exist several proofs of bounds on the enhancement of heat transport due
to convection, quantified with the Nusselt number, $Nu$, with the background
field method (Constantin et al., 1999; Doering & Constantin, 2001; Constantin
et al., 2001; Yan, 2004; Grooms & Whitehead, 2014; Pachev et al., 2020). First
introduced in the 1990s, the background field method provides a tool for
proving bounds on the long-time averages of turbulence (Fantuzzi et al.,
2022). In its original formulation, the idea involves decomposing the flow
variables into a fluctuating and background component satisfying the boundary
conditions to construct a variational problem for bounding the turbulent
dissipation. A bound is obtained by solving the variational problem by
choosing an appropriate background field and using elementary integral
estimates. The method has been successfully used for many fluid flows, none
more so than turbulent convection (Nobili, 2023). Recent insight has shown
that the background method fits within the framework of the auxiliary
functional method (Chernyshenko et al., 2014; Chernyshenko, 2022), which can
yield sharp bounds for well-posed ODEs and PDEs under technical conditions
(Tobasco et al., 2018; Rosa & Temam, 2022).

A fundamental feature of the background field method is to take energy
balances of the governing equations. However, energy identities fail to
capture the effects of rotation. Note that this is not the case for a fluid
driven by rotating boundaries, like in Taylor-Couette flow (Constantin, 1994;
Ding & Marensi, 2019; Kumar, 2022). For convection subject to the Coriolis
force, standard applications of the background field method do not give a
bound on $Nu$ that includes $E$. One path for progress is in the limit of
infinite $Pr$, where the momentum equation simplifies to a forced Stokes flow,
leading to a diagnostic equation between the velocity and temperature,
facilitating better estimates. Notably, without rotation ($E=\infty$), using
the background method, it was proven, up to constants and logarithms, that
$Nu\leq Ra^{1/3}$ (Doering et al., 2006), where $Ra$ is the temperature
difference based Rayleigh number, improving on the bound of $Nu\leq Ra^{1/2}$
valid at arbitrary $Pr$ (Doering & Constantin, 1996). Under rotation
($E<\infty$) at $Pr=\infty$, established results for RBC are illustrated in
fig. 2.

$E$$Nu$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}(Ra\ln\ln
Ra)^{1/3}}$${\color[rgb]{0.84765625,0.3203125,0.09375}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.3203125,0.09375}Ra^{5/12}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}Ra^{2/5}}$$\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}{Ra^{4/11}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}E\,Ra^{2}}$${\color[rgb]{0.84765625,0.3203125,0.09375}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.3203125,0.09375}E^{2}Ra^{2}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}E^{-4/5}Ra^{1/5}}$${\color[rgb]{0,0.498,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.498,0}E\leq
Ra^{-2/3}}$ABCD$\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}{Ra^{\frac{4}{11}}(E^{-1}+1)^{\frac{4}{11}}}$$\color[rgb]{0.84765625,0.3203125,0.09375}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.3203125,0.09375}{E^{4}Ra^{3}}$$\color[rgb]{0,0.498,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.498,0}{E^{2}Ra^{2}}$$E\leq\frac{1}{Ra^{2}}$$\frac{1}{Ra^{2}}\leq
E\leq\frac{1}{Ra^{\frac{1}{4}}}$$\frac{1}{Ra^{\frac{1}{4}}}\leq
E\leq\frac{1}{Ra^{\frac{1}{6}}(\ln\ln
Ra)^{\frac{5}{12}}}$$\frac{1}{Ra^{\frac{1}{6}}(\ln\ln Ra)^{\frac{5}{12}}}\leq
E$$0$$1$ Figure 2: Illustration of the upper bounds with blue lines ( ) on the
Nusselt number, $Nu$ , in terms of the Rayleigh, $Ra$, and Ekman numbers, $E$,
for infinite Prandtl number rotating Rayleigh-Bénard convection, between no-
slip boundaries. All bounds hold up to constants that determine the exact
sizes of the regions, with red text to show the stress-free result in the same
region. The bounds in A are due to Constantin et al. (1999). The green dashed
line ( ) crossing A and B show the upper bound obtained from the non-
hydrostatic quasigeostrophic approximation (Pachev et al., 2020; Grooms &
Whitehead, 2014). In B, the crossover from the bound $Ra^{2/5}$ (Doering &
Constantin, 2001) to $Ra^{4/11}(E^{-1}+1)^{4/11}$ (Yan, 2004), shown by a blue
circle ( ), is given by the values of the constants of both bounds. The same
holds for C with the crossover from $Ra^{4/11}$ (Yan, 2004) to
$E^{-4/5}Ra^{1/5}$ (Constantin et al., 2001) shown by a blue square ( ).
Region D shows the bounds for buoyancy-driven convection (Otto & Seis, 2011;
Whitehead & Doering, 2011b). The transition from C to D is continuous up to
logarithmic corrections (Constantin et al., 1999).

Typically, high $Pr$ is an undesired restriction on the parameter space when
modelling fluid flows, but proving bounds in the limit of $Pr=\infty$ can be
seen as a first step towards establishing bounds valid for all $Pr$. Recent
studies suggest that for any bound proven at infinite $Pr$ in RBC, a semi-
analytic bound for finite $Pr$ can be obtained under specific conditions
(Tilgner, 2022). The results demonstrate that the bounds for finite $Pr$ are,
to highest order, equivalent to the infinite $Pr$ results, with smaller $R,Pr$
and $E$ corrections. The result is unsurprising since bounds at infinite $Pr$
are generally improvements over those obtained for finite $Pr$, and can be
understood as a consequence of the mathematical closeness of the turbulent
attractors of both systems (Wang, 2007).

When rotation dominates over buoyancy, in RBC heuristic arguments suggest that
$Nu\sim E^{3/2}Ra^{2}$, at arbitrary $Pr$ (King et al., 2013; Plumley &
Julien, 2019; Aurnou et al., 2020). Bounds that scale similarly to the
physical arguments in the rapidly rotating regime can be proven when working
with an asymptotic approximation of the governing equations known as the non-
hydrostatic quasigeostrophic (nhQG) equations (Julien et al., 1996, 2016).
Scaling the horizontal length scales by $E^{1/3}$ and adjusting the time
variable yields the nhQG equations that model the limit of rapidly rotating
convection in a plane layer. Applying the background method to the nhQG
equations gives the green bounds in fig. 2 and are, up to constants, $Nu\leq
E^{2}Ra^{2}$, for no-slip conditions (Pachev et al., 2020).

IHC remains less studied in part due to significant differences in the physics
between RBC. Notably, in uniform IHC between isothermal boundaries, the mean
conductive heat flux is zero, rendering the standard definition of the Nusselt
number inapplicable (Goluskin, 2016). In previous works with zero rotation
(Goluskin, 2016), an alternative measure of the turbulent convection is the
mean temperature, $\overline{\left\langle T\right\rangle}$, where angled
brackets $\left\langle\cdot\right\rangle$ denote a volume and overbars a long-
time average. As the flow becomes increasingly turbulent, the temperature
within the domain becomes homogenised, quantified in a lower value of
$\overline{\left\langle T\right\rangle}$, and a higher proxy Nusselt number
defined as $Nu_{p}=1/\overline{\left\langle T\right\rangle}$. An additional
measure of turbulence is $\smash{\overline{\left\langle wT\right\rangle}}$,
quantifying the portion of heat leaving through each boundary,
$\mathcal{F}_{T}$ and $\mathcal{F}_{B}$, due to the effect of convection
(Goluskin & Spiegel, 2012). For a stationary fluid, the heat supplied leaves
the domain symmetrically out of both boundaries to ensure the statistical
stationarity of the solutions. As the thermal forcing increases, convection
carries heat upwards, causing a higher portion of the heat to leave through
the top relative to the bottom boundary (Goluskin & van der Poel, 2016).

In line with previous works on uniform internally heated convection (Goluskin,
2016; Arslan et al., 2021a, b; Kumar et al., 2022; Arslan et al., 2023; Arslan
& Rojas, 2024), the non-dimensional heat flux out of the top and bottom
boundaries is given by

$\displaystyle\mathcal{F}_{T}$
$\displaystyle:=-\overline{\left\langle\partial_{z}T\right\rangle}_{h}|_{z=1}=\frac{1}{2}+\mbox{$\smash{\overline{\left\langle
wT\right\rangle}}$}\quad\text{and}$ (1a) $\displaystyle\mathcal{F}_{B}$
$\displaystyle:=\phantom{-}\overline{\left\langle\partial_{z}T\right\rangle}_{h}|_{z=0}=\frac{1}{2}-\mbox{$\smash{\overline{\left\langle
wT\right\rangle}}$}.$ (1b)

The non-dimensionalisation sets the limits of $\smash{\overline{\left\langle
wT\right\rangle}}$ as $0$ and $\tfrac{1}{2}$, with each limit corresponding to
no convection and infinitely effective convection, respectively. We seek
bounds of the form, $\mathcal{F}_{B}\geq f_{1}(R,E)$ and
$\overline{\left\langle T\right\rangle}\geq f_{2}(R,E)$, where $f_{1}$ and
$f_{2}$ are functions of $R$ and $E$ to be determined for the emergent
quantities of the flow, in different regimes of $E-R$ space. In previous
applications of the background field method to IHC, bounds for
$\overline{\left\langle T\right\rangle}$ are proven with minor adaptation from
the background method as applied to RBC (Lu et al., 2004; Whitehead & Doering,
2011a, 2012). However, in the case of obtaining bounds on
$\smash{\overline{\left\langle wT\right\rangle}}$ and consequently
$\mathcal{F}_{B}$, it has been established that the variational problem
requires a minimum principle on $T$, which states that temperature in the
domain is greater than or equal to zero (Arslan et al., 2021b). The minimum
principle is necessary to obtain lower bounds on $\mathcal{F}_{B}$ that remain
positive as $R$ increases. In the case of no rotation at $Pr=\infty$, the
best-known lower bound on the mean temperature, up to constants, are
$\overline{\left\langle T\right\rangle}\geq(R\ln{R})^{-1/4}$ (Whitehead &
Doering, 2011a) and $\overline{\left\langle T\right\rangle}\geq R^{-5/17}$
(Whitehead & Doering, 2012) for no-slip and stress-free boundaries.
Conversely, the best-known lower bounds on the heat flux out of the domain are
$\mathcal{F}_{B}\geq R^{-2/3}+R^{-1/2}|\ln{(1-R^{-1/3})}|$ and
$\mathcal{F}_{B}\geq R^{-40/29}+R^{-35/29}|\ln{(1-R^{-10/29})}|$ (Arslan &
Rojas, 2024) for the two different kinematic boundary conditions.

In this paper, we prove that for uniform internally heated convection subject
to rotation at an infinite Prandtl number that,

$\mathcal{F}_{B}\gtrsim\begin{dcases}R^{-1}+R^{-4/5}|\ln{(1-R^{-2/5})}|,&E\leq
R^{-2},\\\ R^{-2/3}E^{2/3}+R^{-1/2}E^{1/2}|\ln{(1-R^{-1/3}E^{1/3})}|,&E\leq
E_{0},\\\ R^{-2/3}+R^{-1/2}|\ln{(1-R^{-1/3})}|,&E_{0}\leq E\leq
E_{m},\end{dcases}$ (2)

and

$\overline{\left\langle T\right\rangle}\gtrsim\begin{dcases}R^{-1/3},&E\leq
R^{-2/3},\\\ R^{-1}E^{-1},&E\geq R^{-2/3},\\\ R^{-2/7}E^{2/7},&E\leq E_{1},\\\
R^{-2/7},&E_{1}\leq E\leq E_{m},\end{dcases}$ (3)

where $\gtrsim$ denotes that the bounds hold up to constants determined in
subsequent sections and $E_{0}$, $E_{1}$ and $E_{m}$ are positive constants
determining the regimes of validity in $E-R$ space of each bound. For
notation, $\lVert f\rVert_{p}^{p}=\int^{1}_{0}f^{p}\textrm{d}z$, for
$p<\infty$ and $\lVert f\rVert_{\infty}=\operatorname*{ess\,sup}_{z\in[0,1]}f$
for $p=\infty$, represents the standard $L^{p}$ norm of
$f:[0,1]\rightarrow\mathbb{R}$. The paper is structured as follows. In section
2, we describe the problem setup before discussing the onset of convection in
section 3. Then section 4 proposes heuristic scaling arguments for rotating
IHC, before we prove bounds on $\mathcal{F}_{B}$ in section 5 and on
$\overline{\left\langle T\right\rangle}$ in section 6. Finally, section 7
offers a brief discussion and concluding remarks.

##  2 Setup

We consider a layer of fluid in a rotating frame of reference between two
horizontal plates separated by a distance $d$ and periodic in the horizontal
($x$ and $y$) directions with periods $L_{x}d$ and $L_{y}d$. The fluid has
kinematic viscosity $\nu$, thermal diffusivity $\kappa$, density $\rho$,
specific heat capacity $c_{p}$ and thermal expansion coefficient $\alpha$.
Gravity acts in the negative vertical direction with strength $g$, the fluid
rotates at rate $\Omega$ and is uniformly heated internally at a volumetric
rate $H$.

To non-dimensionalise the problem, we use $d$ as the characteristic length
scale, $d^{2}/\kappa$ as the time scale and $d^{2}H/\kappa\rho c_{p}$ as the
temperature scale (Roberts, 1967). The velocity of the fluid
$\boldsymbol{u}(\boldsymbol{x},t)=u(\boldsymbol{x},t)\mathbf{e}_{1}+v(\boldsymbol{x},t)\mathbf{e}_{2}+w(\boldsymbol{x},t)\mathbf{e}_{3}$
and temperature $T(\boldsymbol{x},t)$ in the non-dimensional domain
$V=[0,L_{x}]\times[0,L_{y}]\times[0,1]$ are governed by the infinite Prandtl
number Boussinesq equations,

$\displaystyle\bnabla\cdot\boldsymbol{u}$ $\displaystyle=0,$ (4a)
$\displaystyle\bnabla p+E^{-1}\mathbf{e}_{3}\times\boldsymbol{u}$
$\displaystyle=\bnabla^{2}\boldsymbol{u}+R\,T\mathbf{e}_{3},$ (4b)
$\displaystyle\partial_{t}T+\boldsymbol{u}\cdot\bnabla T$
$\displaystyle=\bnabla^{2}T+1.$ (4c)

The non-dimensional numbers are the Ekman and Rayleigh numbers, defined as

$E=\frac{\nu}{2\Omega d^{2}},\qquad\text{and}\qquad R=\frac{g\alpha
Hd^{5}}{\rho c_{p}\nu\kappa^{2}}.$ (5)

The boundary conditions are of no-slip and isothermal temperature,
respectively:

$\displaystyle\boldsymbol{u}|_{z=\\{0,1\\}}=0,$ (6a) $\displaystyle
T|_{z=\\{0,1\\}}=0\,.$ (6b)

Figure 1 provides a schematic for the system under consideration. The vertical
component of the curl and double curl of (4b) gives a diagnostic equation
involving the vertical velocity $w$, the vertical vorticity $\zeta$ and
temperature $T$:

$\displaystyle\bnabla^{4}w=E^{-1}\partial_{z}\zeta-R\,\Delta_{h}T,$ (7a)
$\displaystyle\bnabla^{2}\zeta=-E^{-1}\partial_{z}w,$ (7b)

where $\Delta_{h}=\partial^{2}_{x}+\partial^{2}_{y}$ is the horizontal
Laplacian.

The final ingredients are results from Yan (2004) and Constantin et al. (1999)
and a minimum principle on $T$. We state the results as separate lemmas.

######  Lemma 1 (minimum principle)

Suppose $T(\boldsymbol{x},t)$ solves the heat equation (4c) subject to (6b)
where $\boldsymbol{u}$ satisfies $\bnabla\cdot\boldsymbol{u}=0$ and (6a). Let
the negative parts of $T(\boldsymbol{x},t)$ be

$T_{-}(\boldsymbol{x},t):=\max\\{-T(\boldsymbol{x},t),0\\}.$ (8)

Then

$\left\langle|T_{-}(\boldsymbol{x},t)|^{2}\right\rangle\leq\left\langle|T_{-}(\boldsymbol{x},0)|^{2}\right\rangle\
\exp{(-\mu t)},$ (9)

for some $\mu>0$. In particular, if $T(\boldsymbol{x},0)>0$, then
$T_{-}(\boldsymbol{x},0)=0$ and $T(\boldsymbol{x},t)\geq 0~{}\forall t$.

See the appendix A of Arslan et al. (2021b) for a proof.

######  Lemma 2 (Yan (2004))

Let $w_{\boldsymbol{k}},T_{\boldsymbol{k}}:(0,1)\rightarrow\mathbb{R}$,
functions satisfying (7), $\nabla\cdot\boldsymbol{u}=0$, subject to the
velocity boundary conditions (6a). Then,

  * •

For $\boldsymbol{k}\leq 1$

$\lVert w^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}\leq
c_{1}R(1+\tfrac{1}{4}E^{-2})^{1/4}\lVert T_{\boldsymbol{k}}\rVert_{2},$ (10)

where $c_{1}=6^{1/4}$.

  * •

For $\boldsymbol{k}\geq 1$

$\lVert w^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}\leq
c_{2}R\sqrt{\boldsymbol{k}}\lVert
T_{\boldsymbol{k}}\rVert_{2}+c_{2}RE^{-1}\lVert T_{\boldsymbol{k}}\rVert_{2},$
(11)

where
$c_{2}=1+\frac{e^{2}+1}{e^{2}-1}\frac{4\cosh{1}+2\sinh{1}}{-1+\sinh{1}}\sim
64.8734$.

######  Lemma 3 (Constantin et al. (1999))

Let $w,T:V\rightarrow\mathbb{R}$ be horizontally periodic functions such that
they solve (7) subject to $\nabla\cdot\boldsymbol{u}=0$ and the boundary
conditions (6a), then

$\left\langle|\bnabla^{2}w|^{2}\right\rangle+2\left\langle|\bnabla\zeta|^{2}\right\rangle\leq
R^{2}\left\langle|T|^{2}\right\rangle.$ (12)

##  3 Onset of convection

Before proving bounds on the emergent properties of the turbulence
($\mathcal{F}_{B}$ and $\overline{\left\langle T\right\rangle}$), we briefly
discuss the onset of convection for (4). The trivial solution of (4) is found
by taking $\boldsymbol{u}=0$ and considering a steady state where the
temperature is independent of time. The conductive temperature profile,

$T_{c}=\frac{1}{2}z(1-z),$ (13)

represents the transport of heat by conduction alone. In the case where there
is no rotation ($E=\infty$) for the boundary conditions in (6), the system
becomes linearly unstable for all $R>37\,325.2$ (Goluskin, 2016). However,
unlike Rayleigh-Bénard convection when considering (4) in the form

$\frac{\textrm{d}\boldsymbol{s}}{\textrm{d}t}+\mathcal{L}\boldsymbol{s}+\mathcal{N}(\boldsymbol{s},\boldsymbol{s})+\mathcal{P}p=0,$
(14)

where $\mathcal{L}$ and $\mathcal{P}$ are linear operators, $\mathcal{N}$ is a
bilinear operator and $\boldsymbol{s}=(\boldsymbol{u},T)$ the state vector,
the linear operator $\mathcal{L}$ of (4) has a non-zero skew-symmetric
component and nonlinear instability can occur at $R=26\,926.6$, implying that
subcritical convection can occur for internally heated convection (Straughan,
2013; Goluskin, 2016). For $E<\infty$, rotation stabilises the system to
vertical motion and inhibits the onset of convection. For the case of RBC, the
effect of rotation on the onset of convection is well quantified
(Chandrasekhar, 1961). In this section, we demonstrate a result for the effect
of rotation on the Rayleigh number for linear instability, $R_{L}$, in IHC.

###  3.1 Linear stability

The Rayleigh number up to which the flow is linearly stable is identified by
analysing the evolution of perturbations from the linearised system of (4). In
the non-rotating case, for selected thermal boundary conditions, the
marginally stable states are stationary (Davis, 1969; Herron, 2003). When the
flow is subject to rotation, the condition for steady rolls, as opposed to
oscillatory-in-time rolls, at the onset of convection is unknown. For
comparison, in the case of RBC, the onset modes are steady provided $Pr\geq
0.68$, such that assuming a sufficiently large $Pr$ fluid removes the question
of oscillatory motion in the analysis we carry out here. The precise structure
of the motion at the onset should not affect the asymptotic behaviour of
$R_{L}$ for a small Ekman number. It is noteworthy that in the case of the
rotating internally heated fluid sphere, the first convective modes are always
unsteady (Roberts, 1968; Busse, 1970; Jones et al., 2000) and recent evidence
from numerical simulations confirms the existence subcritical convection
(Guervilly & Cardin, 2016; Kaplan et al., 2017).

Taking the setup as described in section 2, we start by decomposing the
temperature field into perturbations from the conductive profile $T_{c}(z)$,

$T(\boldsymbol{x},t)=\vartheta(\boldsymbol{x},t)+T_{c}(z),$ (15)

to obtain the temperature perturbation, $\vartheta$ equation from (4c) and
boundary conditions of

$\displaystyle\partial_{t}\vartheta+\boldsymbol{u}\cdot\bnabla\vartheta=\bnabla^{2}\vartheta+T_{c}^{\prime}w,$
(16a) $\displaystyle\vartheta|_{z=\\{0,1\\}}=0,$ (16b)

where primes denote derivative with respect to $z$. Then, we look at the
marginally stable stationary states of the linearised system of (4) by
considering the $z$ component of the double curl of the momentum equation and
the vertical component of the vorticity equation; we thus have

$\displaystyle\bnabla^{4}w$
$\displaystyle=E^{-1}\partial_{z}\zeta-R\bnabla_{H}^{2}\vartheta\,,$ (17a)
$\displaystyle\bnabla^{2}\zeta$ $\displaystyle=-E^{-1}\partial_{z}w\,,$ (17b)
$\displaystyle\bnabla^{2}\vartheta$ $\displaystyle=T_{c}^{\prime}w.$ (17c)

Given horizontal periodicity, we take a Fourier series expansion in the
horizontal ($x$ and $y$) directions of the form,

$\begin{bmatrix}\vartheta(x,y,z)\\\ \boldsymbol{u}(x,y,z)\\\
\zeta(x,y,z)\end{bmatrix}=\sum_{\boldsymbol{k}}\begin{bmatrix}\hat{\vartheta}_{\boldsymbol{k}}(z)\\\
\hat{\boldsymbol{u}}_{\boldsymbol{k}}(z)\\\
\hat{\zeta}_{\boldsymbol{k}}(z)\end{bmatrix}\textrm{e}^{i(k_{x}x+k_{y}y)}\,,$
(18)

where the sum is over wavevectors $\boldsymbol{k}=(k_{x},k_{y})$ with
magnitude $k=\sqrt{k_{x}^{2}+k_{y}^{2}}$ and hats denote functions of $z$
only. Then, substituting (18) into (17) gives

$\displaystyle(D^{2}-k^{2})^{2}\hat{w}_{\boldsymbol{k}}$
$\displaystyle=E^{-1}\hat{\zeta}^{\prime}_{\boldsymbol{k}}+Rk^{2}\hat{\vartheta}_{\boldsymbol{k}}\,,$
(19a) $\displaystyle(D^{2}-k^{2})\hat{\zeta}_{\boldsymbol{k}}$
$\displaystyle=-E^{-1}\hat{w}^{\prime}_{\boldsymbol{k}}\,,$ (19b)
$\displaystyle(D^{2}-k^{2})\hat{\vartheta}_{\boldsymbol{k}}$
$\displaystyle=T_{c}^{\prime}\hat{w}_{\boldsymbol{k}}\,,$ (19c)

where $D^{2}=\frac{\textrm{d}^{2}}{\textrm{d}z^{2}}$. For no-slip or stress-
free boundary conditions, $\hat{\zeta}_{\boldsymbol{k}}$ and
$\hat{\vartheta}_{\boldsymbol{k}}$ can be eliminated from (19) to give

$(D^{2}-k^{2})^{3}\hat{w}_{\boldsymbol{k}}+E^{-2}\hat{w}^{\prime\prime}_{\boldsymbol{k}}=Rk^{2}T_{c}^{\prime}\hat{w}_{\boldsymbol{k}}.$
(20)

Here, (20) is an eigenvalue problem and can be solved numerically by fixing
$E$ and finding the $R$ at which the first eigenvalue changes sign. The
equivalent problem for RBC is well documented (Chandrasekhar, 1961), however
unlike RBC, the ODE in (20) admits solutions which are hypergeometric
functions because $T_{c}$ in (13) is a non-constant function. As such, even in
the case of stress-free boundary conditions where
$\hat{w}_{\boldsymbol{k}}=\hat{w}^{\prime\prime}_{\boldsymbol{k}}=\hat{w}^{\prime\prime\prime\prime}_{\boldsymbol{k}}=0$
at both boundaries, (20) becomes complicated to solve. Instead, we consider
the asymptotic regime of small $E$ where a simplified form of (20) gives the
desired asymptotic relation between $R_{L}$ and $E$. Following the argument
first presented in Chandrasekhar (1961), we posit that for $R$ close to
$R_{L}$, the wavenumber $k$ tends to infinity so that we retain only terms in
$E$, $R$ and the highest power in $k$ and (20) becomes

$E^{-2}\hat{w}^{\prime\prime}_{\boldsymbol{k}}=k^{2}(R(\tfrac{1}{2}-z)+k^{4})\hat{w}_{\boldsymbol{k}}.$
(21)

Since (21) is of second order, we require only two boundary conditions.
However, in the simplest set of boundary conditions of stress-free boundaries
where $\hat{w}_{\boldsymbol{k}}$ and $\hat{w}_{\boldsymbol{k}}^{\prime\prime}$
are zero, the problem is over-determined with four boundary conditions. It
suffices to take
$\hat{w}_{\boldsymbol{k}}(0)=\hat{w}^{\prime\prime}_{\boldsymbol{k}}(0)=0$,
such that we can make the ansätz that

$\hat{w}_{\boldsymbol{k}}=Ai(n-mz)-\frac{Ai(n)}{Bi(n)}Bi(n-mz),$ (22)

where $Ai(z)$ and $Bi(z)$ are the Airy functions of the first and second kind.
Substituting (22) back into (21) gives

$E^{-2}m^{2}(n-mz)=\frac{1}{2}k^{2}R+k^{6}-k^{2}Rz,$ (23)

from which we require the choice that $m=(k^{2}RE^{2})^{\frac{1}{3}}$.
Substituting for $m$ and rearranging, we obtain

$-\frac{1}{2}E^{2/3}k^{2/3}R+nR^{2/3}-E^{2/3}k^{14/3}=0.$ (24)

Figure 3: Plot of $R_{L}$ against $k$ as given by (25) for Ekman numbers
ranging from $10^{-37}$ ( ) to $10^{-13}$ ( ). The blue dots represent the
minimum $R_{L}$ in $k$, and the inset demonstrates that the minimum wavenumber
$k_{m}$ varies as $E^{-1/3}$ with a compensated plot.

Given that (24) is a cubic equation in $R^{1/3}$, by application of the cubic
formula, we find the real root to be,

$\displaystyle R_{L}^{1/3}$
$\displaystyle=\frac{2n}{3E^{2/3}k^{2/3}}+\frac{1}{3E^{2/3}k^{2/3}}\left(-8n^{3}+27E^{2}k^{6}+27E^{2}k^{3}\sqrt{k^{6}-\frac{16n^{3}}{27E^{2}}}\right)^{\frac{1}{3}}$
$\displaystyle+\frac{4n^{2}}{3E^{2/3}k^{2/3}}\left(-8n^{3}+27E^{2}k^{6}+27E^{2}k^{3}\sqrt{k^{6}-\frac{16n^{3}}{27E^{2}}}\right)^{-\frac{1}{3}}.$
(25)

Then, we want to find the smallest possible $R_{L}$ in (25) by finding the
minimising $k$ by solving $\partial_{k}R_{L}^{1/3}=0$ and substituting back
into (25) to obtain $R_{L}$ as a function of only $E$. In fig. 3 we plot
$R_{L}$ from (25) against $k$ for a wide range of $E$ ($10^{-37}$ to
$10^{-13}$), highlighting the minimum in $k$ found. The inset in fig. 3
demonstrates that the $k_{m}$ varies as $E^{-1/3}$. Given fig. 3, and noting
that in the asymptotic limit of small $E$ we assume large wavenumbers $k$, it
is natural in (25) to take the minimising wavenumber to be

$k_{m}^{6}=\frac{16n^{3}}{27E^{2}},$ (26)

such that the terms in the cube roots are real and positive. Then,
substituting $k_{m}$ back into (25) and the minimal $R_{L}(n,E)$ is achieved
with $n=1$, such that (25) simplifies to

$R_{L}=\frac{(1+2^{2/3}+2^{4/3})^{3}}{9\cdot 2^{1/3}}E^{-4/3}\sim E^{-4/3}.$
(27)

The asymptotic scaling is equivalent to rotating RBC, unsurprising given the
equivalence of the momentum equations, albeit with different prefactors, and
highlights the inhibiting effect of rotation on the Rayleigh number for the
system to become linearly unstable. In the rest of the paper, we will use (27)
to arrive at heuristic scaling arguments and later contextualise the bounds
proven.

##  4 Heuristics scaling arguments

Owing to the lack of data on uniform internally heated convection subject to
rotation, we cannot comment on the sharpness of the bounds we will prove.
Instead, we can use standard physical arguments to determine possible scaling
laws for $\overline{\left\langle T\right\rangle}$ and $\mathcal{F}_{B}$. In
previous studies, the theory of Grossman & Lohse (Grossmann & Lohse, 2000;
Ahlers et al., 2009) has been adapted to uniform and exponentially varying IHC
to determine scaling laws in the non-rotating case (Wang et al., 2020;
Creyssels, 2020, 2021). Here, we will follow the heuristic arguments presented
in Arslan et al. (2021b) that adapt the ideas of marginal stability and
diffusivity free scaling (Malkus, 1954; Priestley, 1954; Spiegel, 1963) and
combine them with arguments for rotation-dominated convection (Stevenson,
1979; King et al., 2009; Ecke & Shishkina, 2023).

The starting point is to suppose that the heat flux out of the domain, defined
in (1), instead can be written as $\mathcal{F}_{B}\sim\overline{\left\langle
T\right\rangle}/\delta$ and $\mathcal{F}_{T}\sim\overline{\left\langle
T\right\rangle}/\varepsilon$, remembering that energy balance gives that
$\mathcal{F}_{B}+\mathcal{F}_{T}=1$. In this section, $\sim$ means
approximately equal to up to constants. Then, in the buoyancy and rotation-
dominated regimes, we suppose that to the highest order, the mean temperature
is a function of the Rayleigh and Ekman numbers, more precisely

$\overline{\left\langle T\right\rangle}\sim
R^{-\alpha},\quad\text{and}\quad\overline{\left\langle
T\right\rangle}\sim(R/R_{L})^{-\gamma}=R^{-\gamma}E^{-4\gamma/3}\,,$ (28)

where $\alpha\in\mathbb{R}_{+}$ and $\gamma\in\mathbb{R}_{+}$ are exponents to
be determined and we have substituted for $R_{L}$ with (27). The two main
regimes of turbulent convection can be interpolated by varying the Rayleigh
and Ekman numbers. However, going from buoyancy to rotation-dominated heat
transport, the thermal boundary layer $\delta$ will become larger than the
Ekman boundary layer $\delta_{E}$. Note that for IHC, the thermal boundary
layers at the boundaries are assumed to be of different widths. Further,
assuming the internal heating balances the diffusion in the boundary layer,
dimensional arguments give that
$\delta\sim\varepsilon^{1/2}\sim\overline{\left\langle T\right\rangle}^{1/2}$,
implying that $\mathcal{F}_{B}\sim\overline{\left\langle
T\right\rangle}^{1/2}$. Whereas for the Ekman boundary layers by standard
arguments $\delta_{E}\sim E^{1/2}$ (Stevenson, 1979). Therefore, using (28)
and supposing $\delta\sim\delta_{E}$ gives

$\gamma=\frac{3\alpha}{3-4\alpha}.$ (29)

The relationship in (29) gives a wide range of possible scaling behaviours for
the IHC as the flow transitions from buoyancy to rotation-dominated, and it
then remains to determine $\alpha$. If we first rearrange (29) in terms of
$\alpha$, then we find that $\alpha=3\gamma/(3+4\gamma)$, meaning that if we
consider $\gamma\rightarrow\infty$, then $\alpha\rightarrow\tfrac{3}{4}$,
implying that the maximal exponent of $\alpha$ is $\tfrac{3}{4}$, though as we
will see this does not correspond to any physical arguments and is ruled out
by rigorous bounds (Lu et al., 2004; Whitehead & Doering, 2011a).

It remains to determine $\alpha$ to obtain the desired heuristic scaling laws.
If we use the argument of marginal stability (Malkus, 1954; Howard, 1963) to
the unstably stratified upper thermal boundary layer, $\varepsilon$, we find
that $\alpha=\tfrac{1}{4}$. If, instead, turbulent heat transport is
independent of the fluid diffusivities and is given by a characteristic free-
fall velocity (Spiegel, 1963), we find $\alpha=\tfrac{1}{3}$. See Arslan et
al. (2021b) for a detailed explanation of the exponents for IHC in the non-
rotating case. Then, for $\alpha=\tfrac{1}{4}$ or $\tfrac{1}{3}$, using (29)
gives the following predictions in the buoyancy and rotation-dominated
regimes,

$\displaystyle\text{non-rotating}:\qquad\overline{\left\langle
T\right\rangle}$
$\displaystyle\sim\begin{dcases}R^{-1/4},&~{}~{}\text{classical},\\\
R^{-1/3},&~{}~{}\text{ultimate},\end{dcases}$ (30a)
$\displaystyle\text{rotating}:\qquad\overline{\left\langle T\right\rangle}$
$\displaystyle\sim\begin{dcases}R^{-3/8}E^{-1/2},&~{}~{}\text{classical},\\\
R^{-3/5}E^{-4/5},&~{}~{}\text{ultimate},\end{dcases}$ (30b)

and

$\displaystyle\text{non-rotating}:\qquad\mathcal{F}_{B}$
$\displaystyle\sim\begin{dcases}R^{-1/8},&~{}~{}\text{classical},\\\
R^{-1/6},&~{}~{}\text{ultimate},\end{dcases}$ (31a)
$\displaystyle\text{rotating}:\qquad\mathcal{F}_{B}$
$\displaystyle\sim\begin{dcases}R^{-3/16}E^{-1/4},&~{}~{}\text{classical},\\\
R^{-3/10}E^{-2/5},&~{}~{}\text{ultimate}.\end{dcases}$ (31b)

While the classical regime for a rotating flow is not physically relevant, it
appears in (30b) and (31b) for completeness.

As mentioned in the introduction, one can define a proxy Nusselt number as
$Nu_{p}=1/\overline{\left\langle T\right\rangle}$. Furthermore, the
temperature difference based Rayleigh number $Ra$ is related to the flux-based
Rayleigh number $R$ through the relation that $Nu_{p}=R/Ra$. Therefore,
substituting for $R$ in the scaling laws (30a) and (30b) returns the known
scaling laws for the Nusselt number in RBC of $Nu\sim Ra^{1/2}$ and $Nu\sim$
$Ra^{3/2}E^{2}$ for the ultimate scaling. We comment on the heuristic scaling
laws in section 7 and compare them to the bounds we prove in the subsequent
sections.

##  5 Bounds on the heat flux out of the domain

In this section, we present proofs of the bounds in (2) on $\mathcal{F}_{B}$
as defined by (1). To obtain a lower bound on $\mathcal{F}_{B}$, we prove
upper bounds on $\smash{\overline{\left\langle wT\right\rangle}}$ by the
background method in the framework of auxiliary functionals (Arslan et al.,
2023). First, in section 5.1, we derive the variational problem for finding
$U$ where $\mbox{$\smash{\overline{\left\langle wT\right\rangle}}$}\leq U$. In
section 5.2, we outline the preliminary choices that are made for the proofs
and in section 5.3, estimate the upper bound on $\smash{\overline{\left\langle
wT\right\rangle}}$. Then, we first prove a bound on
$\smash{\overline{\left\langle wT\right\rangle}}$ valid for large Ekman
numbers $E$ in section 5.4 by the use of lemma 2, followed by a proof valid
for small $E$ in section 5.5 by using lemma 3. To provide an overview, fig. 4
illustrates the lower bounds on $\mathcal{F}_{B}$, omitting the logarithmic
corrections for brevity.

###  5.1 The auxiliary functional method

Here, we outline the main steps in constructing the variational problem to
obtain an upper bound on $\smash{\overline{\left\langle wT\right\rangle}}$.
See previous works for a detailed derivation (Arslan et al., 2021b, 2023). To
prove an upper bound on $\smash{\overline{\left\langle wT\right\rangle}}$, we
employ the auxiliary function method (Chernyshenko et al., 2014; Fantuzzi et
al., 2022). The method relies on the observation that the time derivative of
any bounded and differentiable functional $\mathcal{V}\\{T(t)\\}$ along
solutions of the Boussinesq equations (4) averages to zero over infinite time,
so that

$\overline{\left\langle wT\right\rangle}=\overline{\left\langle
wT\right\rangle+\tfrac{\rm d}{{\rm d}t}\mathcal{V}\\{T(t)\\}}.$ (32)

Two key simplifications follow. The first is that we can estimate (32) by the
pointwise-in-time maximum along the solutions of the governing equations, and
this value is estimated by the maximum it can take over all velocity and
temperature fields that are periodic in $x$ and $y$, satisfying
incompressibility (4a), the boundary conditions (6) and the maximum principle
lemma 1.

$E$$\mathcal{F}_{B}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}E^{2/3}R^{-2/3}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}R^{-2/3}}$${\color[rgb]{0,0.498,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.498,0}R^{-1}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}R^{-2/3}}$${\color[rgb]{0.84765625,0.3203125,0.09375}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.3203125,0.09375}R^{-40/29}}$ABCD$E\leq
R^{-2}$$0$$0$$E_{m}$$E_{0}$$\frac{1}{2}$ Figure 4:  Illustrations of the lower
bounds on the flux out of the bottom boundary, $\mathcal{F}_{B}$, between no-
slip and isothermal boundary conditions at infinite $Pr$, the logarithmic
corrections in the bounds, eq. 2, are suppressed for brevity. The blue plot (
) is the bound derived in section 5.4, valid up to $E=E_{m}\sim 41.4487$ and
$E_{0}=4.1688$. The bound derived in section 5.5 is shown with a green line (
). The dashed horizontal line denotes when all heat transport is by conduction
and $\mathcal{F}_{B}=\frac{1}{2}$. In D ( ), we show the bounds for stress-
free (top, red) and no-slip (bottom, blue) at zero rotation (Arslan & Rojas,
2024).

We restrict our attention to quadratic functionals taking the form

$\mathcal{V}\\{T\\}:=\left\langle\frac{\beta}{2}|T|^{2}-[\tau(z)+z-1]T\right\rangle,$
(33)

that are parametrised by a positive constant $\beta>0$, referred to as the
balance parameter and a piecewise-differentiable function
$\tau:[0,1]\to\mathbb{R}$ with a square-integrable derivative that we call the
background temperature field. Here $\tau(z)$ satisfies

$\tau(0)=1,\qquad\tau(1)=0.$ (34)

Introducing a constant, $U$, and rearranging, (32) can be written as,

$\mbox{$\smash{\overline{\left\langle wT\right\rangle}}$}\leq U-U+\left\langle
wT\right\rangle+\tfrac{\textrm{d}}{\textrm{d}t}\mathcal{V}\\{T\\}\leq U,$ (35)

where the final inequality holds given that, $U-\left\langle
wT\right\rangle-\tfrac{\textrm{d}}{\textrm{d}t}\mathcal{V}\\{T\\}\geq 0$,
where we can substitute for the Lie derivative of $\mathcal{V}\\{T\\}$ by
using (4c). However, the minimum principle, lemma 1, is necessary to obtain a
$R$-dependent bound on $\smash{\overline{\left\langle wT\right\rangle}}$ that
approaches $\tfrac{1}{2}$ from below as $R$ increases. The condition is
enforced with a Lagrange multiplier, $\lambda(z)$, so that the problem
statement after computations as outlined in previous work (Arslan et al.,
2021b, 2023) becomes

$\displaystyle\mbox{$\smash{\overline{\left\langle
wT\right\rangle}}$}\leq\inf_{U,\beta,\tau(z),\lambda(z)}\\{U~{}|~{}\mathcal{S}\\{\boldsymbol{u},T\\}\geq
0\quad\forall(\boldsymbol{u},T)\in\mathcal{H}_{+}\\},$ (36)

where

$\mathcal{H}_{+}=\\{(\boldsymbol{u},T)|\text{horizontally
periodic},\bnabla\cdot\boldsymbol{u}=0,\eqref{bc:non_uni},\eqref{eq:mom_eqs_diag},T(\boldsymbol{x})\geq
0~{}~{}\text{a.e.}~{}~{}\boldsymbol{x}\in V\\},$

provided $\lambda(z)$ is a non-decreasing function, and

$\mathcal{S}\\{\boldsymbol{u},T\\}:=\left\langle\beta|\nabla
T|^{2}+\tau^{\prime}(z)wT+(\beta
z-\tau^{\prime}(z)+\lambda(z))\partial_{z}T+\tau(z)+U\right\rangle-\tfrac{1}{2}.$
(37)

Ensuring the positivity of the quadratic terms in (37) is referred to as the
spectral constraint and is defined as

$\left\langle\beta|\bnabla T|^{2}+\tau^{\prime}(z)wT\right\rangle\geq 0\,,$
(38)

where $w$ and $T$ are related by (7) and subject to the boundary conditions
(6). As has been previously established (Arslan et al., 2023), provided the
spectral constraint is satisfied, then the non-negativity of
$\mathcal{S}\\{\boldsymbol{u},T\\}$ is ensured when $U$ is given by

$\mbox{$\smash{\overline{\left\langle wT\right\rangle}}$}\leq
U:=\frac{1}{2}+\inf_{\begin{subarray}{c}\lambda\in L^{2}(0,1)\\\
\lambda~{}\textrm{non decreasing}\\\
\left\langle\lambda\right\rangle=-1\end{subarray}}\left\langle\frac{1}{4\beta}\left|\beta\left(z-\frac{1}{2}\right)-\tau^{\prime}(z)+\lambda(z)\right|^{2}-\tau(z)\right\rangle.$
(39)

###  5.2 Preliminaries

To establish a bound on $\smash{\overline{\left\langle wT\right\rangle}}$, we
state the main choices used in the proof that minimise $U(\beta,\tau,\lambda)$
as defined in (39). We make the following choice of background temperature
field,

$\tau(z):=\begin{dcases}1-\frac{1-a}{\delta}z,&0\leq z\leq\delta,\\\
a,&\delta\leq z\leq 1-\varepsilon,\\\
\frac{a(1-\varepsilon)}{\varepsilon}\left(\frac{1-z}{z}\right),&1-\varepsilon\leq
z\leq 1,\end{dcases}$ (40)

and set $\lambda(z)$ to be

$\lambda(z):=\begin{dcases}-\frac{1-a}{\delta},&0\leq z\leq\delta,\\\
-\frac{a}{1-\delta},&\delta\leq z\leq 1.\end{dcases}$ (41)

The piecewise functions $\tau(z)$ and $\lambda(z)$ are quantified by the
boundary layer widths $\delta\in(0,\tfrac{1}{3})$ and
$\varepsilon\in(0,\tfrac{1}{3})$, where $\delta\leq\varepsilon$, and parameter
$a>0$ that determines the value of $\tau(z)$ in the bulk. See fig. 5 for a
sketch of the functions.

$z$$\tau(z)$$1$$1$$\delta$$1-\varepsilon$$z$$\lambda(z)$$1$$\delta$$-\frac{1-a}{\delta}$
Figure 5: Sketches of the functions $\tau(z)$ in (40) and $\lambda(z)$ in (41)
used to prove (2), where $\delta$ is the boundary layer width at the bottom,
$\varepsilon$ the boundary layer width at the top of the domain and $a$ is eq.
42.

We will further fix

$a=\frac{1}{2}\delta\varepsilon^{1/2},$ (42)

and

$\beta=\frac{\left\langle|\tau^{\prime}(z)-\lambda(z)|^{2}\right\rangle^{1/2}}{\left\langle|z-\tfrac{1}{2}|^{2}\right\rangle^{1/2}}.$
(43)

In the following subsections, we prove bounds for different regimes of the
Ekman number. We achieve this by using different estimates on the spectral
constraint (38). However, the expression for the upper bound on
$\smash{\overline{\left\langle wT\right\rangle}}$ in (39) remains the same.
Therefore, first, we use our choices of $\tau(z)$ in (40) and $\lambda(z)$ in
(41) to estimate (39).

###  5.3 Estimating the upper bound

Starting with (39) an application of the triangle inequality and the choice of
$\beta$ in (43) gives

$U\leq\frac{1}{2}+\frac{1}{\sqrt{12}}\left\langle|\tau^{\prime}(z)-\lambda(z)|^{2}\right\rangle^{1/2}-\left\langle\tau\right\rangle.$
(44)

Then, evaluating the sign positive integral with $\tau(z)$ from (40) and
$\lambda(z)$ in (41), gives

$\displaystyle\left\langle|\tau^{\prime}(z)-\lambda(z)|^{2}\right\rangle$
$\displaystyle=\int^{1}_{\delta}|\tau^{\prime}(z)-\lambda(z)|^{2}\textrm{d}z$
$\displaystyle=\frac{a^{2}}{(1-\delta)^{2}}(1-\varepsilon-\delta)+\frac{a^{2}}{\varepsilon^{2}}\int^{1}_{1-\varepsilon}\left(\frac{1-\varepsilon}{z^{2}}-\frac{\varepsilon}{1-\delta}\right)^{2}\textrm{d}z\,.$
(45)

We will require an upper and lower bound on (45). Starting with a lower bound,
given that $\varepsilon\leq\frac{1}{3}$ and $\delta\leq\frac{1}{3}$, with $z$
in the range $(1-\varepsilon,1)$, we make the suboptimal but simple estimate
that

$\left(\frac{1-\varepsilon}{z^{2}}-\frac{\varepsilon}{1-\delta}\right)^{2}\geq\frac{1}{9},$

such that we get

$\displaystyle\left\langle|\tau^{\prime}(z)-\lambda(z)|^{2}\right\rangle\geq\frac{a^{2}}{\varepsilon^{2}}\int^{1}_{1-\varepsilon}\left(\frac{(1-\varepsilon)}{z^{2}}-\frac{\varepsilon}{1-\delta}\right)^{2}\textrm{d}z\geq\frac{a^{2}}{9\varepsilon}.$
(46)

For an upper bound on (45), given that $\varepsilon$ and $\delta$ are
positive, bounded above by $\frac{1}{3}$ and that $\delta\leq\varepsilon$, we
use the estimate
$(1-\varepsilon-\delta)(1-\delta)^{-2}\leq\frac{3}{4}\varepsilon^{-1}$ and
$z^{-2}\leq(1-\varepsilon)^{-2}$, to obtain

$\displaystyle\left\langle|\tau^{\prime}(z)-\lambda(z)|^{2}\right\rangle$
$\displaystyle\leq\frac{3a^{2}}{4\varepsilon}+\frac{a^{2}}{\varepsilon^{2}}\int^{1}_{1-\varepsilon}\left(\frac{1-\varepsilon}{z^{2}}-\frac{\varepsilon}{1-\delta}\right)^{2}\textrm{d}z$
$\displaystyle\leq\frac{3a^{2}}{4\varepsilon}+\frac{a^{2}}{\varepsilon^{2}}\int^{1}_{1-\varepsilon}\frac{1}{(1-\varepsilon)^{2}}+\frac{\varepsilon^{2}}{(1-\delta)^{2}}\textrm{d}z$
$\displaystyle\leq\frac{3a^{2}}{4\varepsilon}+\frac{a^{2}}{\varepsilon}\frac{1+\varepsilon^{2}}{(1-\varepsilon)^{2}}\leq\frac{4a^{2}}{\varepsilon}.$
(47)

Moving on to the integral of $\tau(z)$ in (39), we have that

$\int^{1}_{0}\tau(z)~{}\textrm{d}z=\frac{1}{2}\delta(1-a)-\frac{a}{\varepsilon}(1-\varepsilon)\ln(1-\varepsilon).$
(48)

Substituting (48) and (47) back into (44), taking $a$ as given by (42) and
$\varepsilon,\delta\leq\frac{1}{3}$ such that
$\delta^{2}\varepsilon^{1/2}\leq\frac{\sqrt{3}}{9}\delta$, gives

$U\leq\frac{1}{2}-n\delta-\frac{1}{3}\delta\varepsilon^{-1/2}|\ln{(1-\varepsilon)}|\,,$
(49)

where $n=\frac{18-7\sqrt{3}}{36}$.

###  5.4 Large Ekman numbers

To obtain bounds for large $E$ in this subsection, we will use lemma 2. The
estimates in lemma 2 are pointwise estimates of the vertical velocity in
Fourier space. Therefore, we exploit the horizontal periodicity of
$\boldsymbol{u}$ and $T$ and take a Fourier decomposition of $w$ and $T$ in
the spectral constraint (38). Taking that

$\begin{bmatrix}T(x,y,z)\\\
\boldsymbol{u}(x,y,z)\end{bmatrix}=\sum_{\boldsymbol{k}}\begin{bmatrix}T_{\boldsymbol{k}}(z)\\\
\boldsymbol{u}_{\boldsymbol{k}}(z)\end{bmatrix}\textrm{e}^{i(k_{x}x+k_{y}y)}\,,$
(50)

where the sum is over nonzero wavevectors $\boldsymbol{k}=(k_{x},k_{y})$ for
the horizontal periods $L_{x}$ and $L_{y}$ and magnitude of each wavevector is
$k=\sqrt{k_{\smash{x}}^{2}+k_{\smash{y}}^{2}}$. To lighten the notation in
what follows, unlike in section 3, we do not use hats for the Fourier
coefficients, even though the functions represent the same quantities.
Inserting the Fourier expansions (50) into (38) give

$\int^{1}_{0}\beta|{T}^{\prime}_{\boldsymbol{k}}|^{2}+\beta
k^{2}|{T}_{\boldsymbol{k}}|^{2}+\tau^{\prime}(z)|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\geq
0,$ (51)

where the complex conjugate relations of
${w}_{\boldsymbol{k}}={w}^{*}_{\boldsymbol{k}}$ holds, and
$w_{\boldsymbol{k}}$ and $T_{\boldsymbol{k}}$ are subject to the boundary
conditions

$\displaystyle{w}_{\boldsymbol{k}}(0)={w}_{\boldsymbol{k}}(1)={w}^{\prime}_{\boldsymbol{k}}(0)={w}^{\prime}_{\boldsymbol{k}}(1)=0,$
(52a) $\displaystyle{T}_{\boldsymbol{k}}(0)={T}_{\boldsymbol{k}}(1)=0.$ (52b)

Based on the boundary conditions, we infer the following two estimates. Given
(52a) applying the fundamental theorem of calculus and Hölders inequality
gives

$|{w}_{\boldsymbol{k}}|=\int^{z}_{0}\int^{\sigma}_{0}|\partial^{2}_{\eta}{w}_{\boldsymbol{k}}(\eta)|\textrm{d}\eta\textrm{d}\sigma\leq\frac{1}{2}z^{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty},$
(53)

and for ${T}_{\boldsymbol{k}}$, the fundamental theorem of calculus and the
Cauchy-Schwarz inequality gives

$|{T}_{\boldsymbol{k}}|=\int^{z}_{0}|\partial_{\eta}{T}_{\boldsymbol{k}}(\eta)|\textrm{d}\eta\leq\sqrt{z}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}.$
(54)

Next, we substitute (40) into the sign indefinite term in (51) to obtain

$\int^{1}_{0}\tau^{\prime}(z)|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z=-\frac{1-a}{\delta}\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z-\frac{a(1-\varepsilon)}{\varepsilon}\int^{1}_{1-\varepsilon}z^{-2}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z.$
(55)

As lemma 2 contains two estimates for different regimes of $\boldsymbol{k}$,
we will split the sign indefinite term in half. Then, given that $a\leq 1$,
use of (53), the Cauchy-Schwarz inequality, (10) and (11) from lemma 2 gives
that

$\displaystyle\frac{1-a}{\delta}\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z$
$\displaystyle\leq\frac{1}{2\delta}\int^{\delta}_{0}z^{5/2}\textrm{d}z\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}$
$\displaystyle\leq\frac{\delta^{5/2}}{14}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}+\frac{\delta^{5/2}}{14}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}$
$\displaystyle\leq\frac{c_{1}}{14}\delta^{5/2}R(1+\tfrac{1}{4}E^{-2})^{1/4}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}$
$\displaystyle\quad+\frac{c_{2}}{14}\delta^{5/2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\Big{(}R\sqrt{k}\lVert{T}_{\boldsymbol{k}}\rVert_{2}+RE^{-1}\lVert{T}_{\boldsymbol{k}}\rVert_{2}\Big{)}\,.$
(56)

Taking the term of order $\sqrt{k}$ in (56), we estimate further by noting
that from (54) we have a standard Poincaré inequality of

$\lVert{T}_{\boldsymbol{k}}\rVert_{2}\leq\frac{1}{\sqrt{2}}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2},$
(57)

such that the use of Youngs’ inequality twice gives

$\displaystyle\frac{1}{14}\delta^{5/2}R\sqrt{k}c_{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}$
$\displaystyle\leq\frac{\beta}{2}k\lVert{T}_{\boldsymbol{k}}\rVert_{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}+\frac{c_{2}^{2}}{392\beta}\delta^{5}R^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}$
$\displaystyle\leq\frac{\beta}{2}k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{\beta}{8}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}+\frac{c_{2}^{2}}{392\sqrt{2}\beta}\delta^{5}R^{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(58)

Then, substituting (58) into (56), the integral at the lower boundary becomes

$\displaystyle\frac{1-a}{\delta}\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\leq$
$\displaystyle\frac{\beta}{2}k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{\beta}{8}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{c_{2}^{2}}{392\sqrt{2}\beta}\delta^{5}R^{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}$
$\displaystyle+\frac{c_{1}}{14\sqrt{2}}\delta^{5/2}R(1+\tfrac{1}{4}E^{-2})^{1/4}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{c_{2}}{14\sqrt{2}}\delta^{5/2}RE^{-1}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(59)

We realise that for a sufficiently small Ekman number, the term of order
$E^{-1}$ is larger than $(1+E^{-2})^{1/4}$ such that if we make the estimate

$c_{1}(1+\tfrac{1}{4}E^{-2})^{1/4}\leq c_{2}E^{-1},$ (60)

we get a quadratic form in terms of $E^{2}$, that places an upper bound on $E$
of

$E\leq
E_{m}=\frac{1}{2\sqrt{2}}\left(-1+\sqrt{1+64(c_{2}^{4}/c_{1}^{4})}\right)^{1/2}=41.4487.$
(61)

Now, (59) becomes,

$\displaystyle\frac{1-a}{\delta}\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\leq$
$\displaystyle\frac{\beta}{2}k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{\beta}{8}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{c_{2}^{2}}{392\sqrt{2}\beta}\delta^{5}R^{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{c_{2}}{7\sqrt{2}}\delta^{5/2}RE^{-1}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(62)

Returning to the integral at the upper boundary in (55), we apply the same
procedure, where (53) and (54) are instead

$|{w}_{\boldsymbol{k}}|\leq\frac{1}{2}(1-z)^{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty},\qquad|{T}_{\boldsymbol{k}}|\leq\sqrt{1-z}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}.$
(63)

Given $\varepsilon\leq\frac{1}{3}$ we use that
$z^{-2}\leq(1-\varepsilon)^{-2}$ to get

$\displaystyle\frac{a(1-\varepsilon)}{\varepsilon}\int^{1}_{1-\varepsilon}z^{-2}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\leq\,$
$\displaystyle\frac{a}{\varepsilon(1-\varepsilon)}\int^{1}_{1-\varepsilon}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\leq\frac{3a}{2\varepsilon}\int^{1}_{1-\varepsilon}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z.$
(64)

By use of lemma 2, along with (63), (57), Youngs’ inequality, and (60), we can
estimate the integral at the upper boundary to obtain

$\displaystyle\frac{a(1-\varepsilon)}{\varepsilon}\int^{1}_{1-\varepsilon}z^{-2}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z$
$\displaystyle\leq\frac{3a}{2\varepsilon}\int^{1}_{1-\varepsilon}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\leq\,\frac{\beta}{2}k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{\beta}{8}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}$
$\displaystyle+\frac{9c_{2}^{2}}{6272\sqrt{2}\beta}\delta^{2}\varepsilon^{6}R^{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{3}{28\sqrt{2}}c_{2}\delta\varepsilon^{3}RE^{-1}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(65)

Substituting (62) and (65) back into the spectral constraint (51) gives

$\displaystyle\left(\frac{3\beta}{4}-\frac{c_{2}^{2}}{392\sqrt{2}}\frac{\delta^{5}R^{2}}{\beta}-\frac{c_{2}}{7\sqrt{2}}\delta^{5/2}RE^{-1}-\frac{3c_{2}}{28\sqrt{2}}\delta\varepsilon^{3}RE^{-1}-\frac{9c_{2}^{2}}{6272\sqrt{2}}\frac{\delta^{2}\varepsilon^{6}R^{2}}{\beta}\right)\lVert
T^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}\geq 0.$ (66)

The spectral condition is satisfied provided the term in the brackets of (66)
is non-negative. Note that we have an explicit expression for $\beta$ in (43),
which in conjunction with the lower bound in (46) gives the following lower
bound on $\beta$ of

$\beta\geq\frac{\sqrt{3}}{3}\delta.$ (67)

After estimating $\beta$ from below with (67) and making the choice

$\delta=\left(\frac{9}{16}\right)^{1/3}\varepsilon^{2},$ (68)

the condition for the positivity of (66) becomes, after rearranging,

$1-\frac{c_{2}^{2}}{49\sqrt{2}}\delta^{3}R^{2}-\frac{8\,c_{2}}{7\sqrt{6}}\delta^{3/2}RE^{-1}\geq
0.$ (69)

In (69), two possible choices of $\delta=\delta(R,E)$ guarantee the non-
negativity of the left-hand side. If the second negative term dominates the
first, i.e.

$\frac{c_{2}^{2}}{49\sqrt{2}}\delta^{3}R^{2}\leq\frac{8\,c_{2}}{7\sqrt{6}}\delta^{3/2}RE^{-1},$

then (69) becomes

$\delta\leq\left(\frac{7\sqrt{6}}{16c_{2}}\right)^{2/3}R^{-2/3}E^{2/3}.$ (70)

Taking $\delta$ as large as possible in (70) and substituting back into (69)
implies that $E\leq 8(\sqrt{2}/3)^{1/2}$. Whereas in the opposite scenario
where,

$\frac{8\,c_{2}}{7\sqrt{6}}\delta^{3/2}RE^{-1}\leq\frac{c_{2}^{2}}{49\sqrt{2}}\delta^{3}R^{2},$

then (69) becomes

$\delta\leq\left(\frac{49\sqrt{2}}{2c_{2}^{2}}\right)^{1/3}R^{-2/3},$ (71)

which holds for $E\geq 8(\sqrt{2}/3)^{1/2}$. In summary, the spectral
condition holds if the condition in (69) is satisfied and (69) is guaranteed
when we take $\delta$ as large as possible in (70) and (71). As a result, we
have that

$\delta=\begin{dcases}\left(\frac{7\sqrt{6}}{16c_{2}}\right)^{2/3}R^{-2/3}E^{2/3},&E\leq
8(\sqrt{2}/3)^{1/2},\\\
\left(\frac{49\sqrt{2}}{2c_{2}^{2}}\right)^{1/3}R^{-2/3},&E\geq
8(\sqrt{2}/3)^{1/2}.\end{dcases}$ (72)

and by (68) that,

$\varepsilon=\begin{dcases}\left(\frac{7\sqrt{6}}{12c_{2}}\right)^{1/3}R^{-1/3}E^{1/3},&E\leq
8(\sqrt{2}/3)^{1/2},\\\
\left(\frac{392\sqrt{2}}{9c_{2}^{2}}\right)^{1/6}R^{-1/3},&E\geq
8(\sqrt{2}/3)^{1/2}.\end{dcases}$ (73)

Therefore, substituting (72) and (73) back into (49), along with the fact that
$\mbox{$\smash{\overline{\left\langle wT\right\rangle}}$}\leq U$ and (1) to
obtain

$\mathcal{F}_{B}\geq\begin{dcases}d_{1}R^{-2/3}E^{2/3}+d_{2}R^{-1/2}E^{1/2}|\ln(1-d_{3}R^{-1/3}E^{1/3})|,&E\leq
8(\sqrt{2}/3)^{1/2},\\\
d_{4}R^{-2/3}+d_{5}R^{-1/2}|\ln(1-d_{6}R^{-1/3})|,&8(\sqrt{2}/3)^{1/2}\leq
E\leq E_{m},\end{dcases}$ (74)

where the constants $d_{1}$ to $d_{6}$ are collated in appendix A. Finally, in
section 5.2 we chose that both boundary layer widths are in $(0,\frac{1}{3})$,
therefore given (72) and (73) the bound obtained in (74) holds for all $R\geq
0.4715$.

###  5.5 Small Ekman numbers

Next, we demonstrate a proof of the bound on $\mathcal{F}_{B}$ valid for small
$E$ in (2). In this regime, we use lemma 3 to demonstrate the non-negativity
of the spectral constraint (38). The estimates used in this subsection do not
require estimates in Fourier space.

Starting with the spectral constraint in (38), we substitute $\tau(z)$ from
(40) into the sign-indefinite term to obtain

$\left\langle\tau^{\prime}(z)wT\right\rangle=-\frac{1-a}{\delta}\left\langle\int^{\delta}_{0}wT\textrm{d}z\right\rangle_{h}-\frac{a}{\varepsilon}(1-\varepsilon)\left\langle\int^{1}_{1-\varepsilon}wT\textrm{d}z\right\rangle_{h}\,.$
(75)

We first consider the integral in (75) near $z=0$ and obtain an estimate on
$wT$. Since we require a lower bound on the right-hand side of (75), we can
rearrange the order of integration of the first term on the right-hand side of
(75) and estimate the integral from above. Given the boundary conditions (6),
use of the fundamental theorem of calculus, (7b) and integration by parts
gives

$\displaystyle\langle|wT|\rangle_{h}$
$\displaystyle=\left\langle\left|\int^{z}_{0}\partial_{s}(wT)\textrm{d}s\right|\right\rangle_{h}=\left\langle\left|\int^{z}_{0}T\partial_{s}w+w\partial_{s}T\textrm{d}s\right|\right\rangle_{h}=\left\langle\left|\int^{z}_{0}-ET\bnabla^{2}\zeta+w\partial_{s}T\textrm{d}s\right|\right\rangle_{h}$
$\displaystyle=E\langle|\zeta^{\prime}T|\rangle_{h}+\left\langle\left|\int^{z}_{0}E\bnabla\zeta\bnabla
T+w\partial_{s}T\textrm{d}s\right|\right\rangle_{h}.$ (76)

Then, given the boundary condition on the velocity and temperature in (6), we
have that

$|w(\cdot,z)|\leq\frac{2}{3}z^{3/2}\left(\int^{1}_{0}|w^{\prime\prime}(\cdot,z)|^{2}\textrm{d}z\right)^{1/2},\quad\langle
T^{2}\rangle_{h}^{1/2}\leq\sqrt{z}\left\langle|\bnabla
T|^{2}\right\rangle^{1/2},$ (77)

use of which, along with multiple applications of the Cauchy-Schwarz
inequality in (76), and that for any $f\in L^{2}(0,1)$ we have
$\langle|f^{\prime}|^{2}\rangle_{h}\leq\langle|\bnabla
f|^{2}\rangle_{h}\leq\left\langle|\bnabla f|^{2}\right\rangle$, gives

$\displaystyle\langle|wT|\rangle_{h}$ $\displaystyle\leq
E\langle|\zeta^{\prime}|^{2}\rangle_{h}^{1/2}\langle|T|^{2}\rangle_{h}^{1/2}+E\langle|\bnabla\zeta|^{2}\rangle_{h}^{1/2}\langle|\bnabla
T|^{2}\rangle_{h}^{1/2}+\frac{z^{2}}{3}\left\langle\int^{z}_{0}|\partial_{s}T|^{2}\textrm{d}s\right\rangle^{1/2}_{h}\left\langle|w^{\prime\prime}|^{2}\right\rangle^{1/2}$
$\displaystyle\leq\left((\sqrt{z}+1)E\left\langle|\bnabla\zeta|^{2}\right\rangle^{1/2}+\frac{1}{3}z^{2}\left\langle|\bnabla^{2}w|^{2}\right\rangle^{1/2}\right)\left\langle|\bnabla
T|^{2}\right\rangle^{1/2}\,.$ (78)

Next, we use lemma 3, to bound both
$\left\langle|\bnabla\zeta|^{2}\right\rangle$ and
$\left\langle|\bnabla^{2}w|^{2}\right\rangle$ from above and given that $T$ is
horizontally periodic with Dirichlet boundary conditions at $z=0$ and $1$ we
have the standard Poincaré inequality
$\left\langle|T|^{2}\right\rangle\leq(1/\pi^{2})\left\langle|\bnabla
T|^{2}\right\rangle$, which gives

$\langle|wT|\rangle_{h}\leq\left(\frac{E\,R}{\sqrt{2}\,\pi}\left(1+\sqrt{z}\right)+\frac{1}{3\pi}z^{2}R\right)\left\langle|\bnabla
T|^{2}\right\rangle.$ (79)

Then, substituting back into the integral at the boundary, whereby (42) we
have $1-a\leq 1$, we get that

$\frac{1-a}{\delta}\int^{\delta}_{0}\langle|wT|\rangle_{h}\textrm{d}z\leq\left(\frac{E\,R}{\sqrt{2}\,\pi}\left(1+\frac{2}{3}\delta^{1/2}\right)+\frac{1}{9\pi}\delta^{2}R\right)\left\langle|\bnabla
T|^{2}\right\rangle.$ (80)

The same estimates can be carried out at the upper boundary with $a$ given by
(42) and $z$ replaced by $1-z$. Then, since $\varepsilon\leq\frac{1}{3}$ we
can take $1-\varepsilon\leq 1$ to obtain,

$\frac{1}{2}\delta\varepsilon^{-1/2}(1-\varepsilon)\int^{1}_{1-\varepsilon}\langle|wT|\rangle_{h}\textrm{d}z\leq\frac{\delta\varepsilon^{1/2}}{2}\left(\frac{E\,R}{\sqrt{2}\,\pi}\left(1+\frac{2}{3}\varepsilon^{1/2}\right)+\frac{1}{9\pi}\varepsilon^{2}R\right)\left\langle|\bnabla
T|^{2}\right\rangle.$ (81)

Substituting (80) and (81) back into (75) and then into (38), gives after use
of the lower bound on $\beta$ from (67) that

$\left(\frac{\sqrt{3}\delta}{3}-\frac{E\,R}{\sqrt{2}\pi}\left(1+\frac{2\delta^{1/2}}{3}\right)-\frac{\delta^{2}R}{9\pi}-\frac{\delta\varepsilon^{1/2}E\,R}{\sqrt{2}\pi}\left(1+\frac{2\varepsilon^{1/2}}{3}\right)-\frac{\delta\varepsilon^{5/2}R}{9\pi}\right)\left\langle|\bnabla
T|^{2}\right\rangle\geq 0.$ (82)

Since $\varepsilon\leq\frac{1}{3}$ and $\delta\leq\frac{1}{3}$, then, we will
make estimates, $1+2\delta^{1/2}/3\leq\sqrt{2}\pi/3$ and
$1+2\varepsilon^{1/2}/3\leq\sqrt{2}\pi/3$, such that after rearranging the
spectral constraint becomes

$\sqrt{3}\delta-
ER-\frac{\delta^{2}R}{3\pi}-\delta\varepsilon^{1/2}ER-\frac{\delta\varepsilon^{5/2}R}{3\pi}\geq
0.$ (83)

In (83), the first negative term does not contain an explicit $\delta$
dependence, and so we will at the very least require that that

$ER\leq p\delta,$ (84)

where $p=\sqrt{3}-\frac{3}{2}$ is chosen for algebraic simplicity. Using (84)
and further making the choice

$\delta=\varepsilon^{5/2},$ (85)

gives

$\frac{3}{2}-\frac{2}{3\pi}\delta R-\delta^{1/5}ER\geq 0.$ (86)

Similar to the proof of a bound for large Ekman numbers in section 5.4, we
have a constraint that we consider in two regimes. If in (86), the second
negative term dominates the first such that

$\frac{2}{3\pi}\delta R\leq\delta^{1/5}ER,$

then

$\delta\leq\frac{243}{1024}E^{-5}R^{-5}\,,$ (87)

however, if this $\delta$ is to satisfy the spectral constraint then its
implication on (84) and (86) need to be checked. If we take $\delta$ to be as
large as possible in (87), then, up to constants, we get from (84) and (86)
that

$E\gtrsim R^{-4/5}\quad\textrm{and}\quad E\lesssim R^{-1},$

which leads to a contradiction such that the initial assumption cannot be
true. Assuming instead that,

$\delta^{1/5}ER\leq\frac{2}{3\pi}\delta R,$

gives in (86) that

$\delta\leq\frac{9\pi}{8}R^{-1}.$ (88)

Taking $\delta$ as large as possible in (88), the constraints from (84) and
(86) imply that,

$E\leq\frac{9\pi}{8}(\sqrt{3}-\tfrac{3}{2})R^{-2},\quad\textrm{and}\quad
E\leq\frac{3}{4}\left(\frac{8}{9\pi}\right)^{1/5}R^{-4/5}.$ (89)

The first condition in (89), due to (84), is stronger for all $R>1$ and
defines the domain of validity for the bound. Finally, substituting the
largest $\delta$ from (88) into (85) gives

$\varepsilon=\left(\frac{9\pi}{8}\right)^{2/5}R^{-2/5}.$ (90)

Then, taking $\delta$ as large as possible in (88), (90), substituting into
(49), along with the fact that $\mbox{$\smash{\overline{\left\langle
wT\right\rangle}}$}\leq U$ and (1) we get that,

$\mathcal{F}_{B}\geq
d_{7}R^{-1}+d_{8}R^{-4/5}|\ln{(1-d_{9}R^{-2/5})}|\,,\qquad
E\leq\frac{9\pi}{8}(\sqrt{3}-\tfrac{3}{2})R^{-2},$ (91)

where the constants $d_{7}$ to $d_{9}$ are collated in appendix A. Finally,
for completeness, we verify that the choice of
$\varepsilon,\delta\in(0,\frac{1}{3})$ made in section 5.2 is not restrictive.
Since $\delta$ is smaller than $\varepsilon$ as expressed in eq. 85, taking
$\delta$ as large as possible in (88) the bound obtained in (91) holds for all
$R\geq 10.6029$.

##  6 Bounds on the mean temperature

In this section, we prove bounds on $\overline{\left\langle T\right\rangle}$
with the auxiliary functional method by using the same strategy as in section
5.1. First, we derive a variational problem for obtaining a lower bound on
$\overline{\left\langle T\right\rangle}$, with a different quadratic auxiliary
functional to (33). Then, given our choice of background profile $\varphi(z)$,
we can estimate the lower bound $L$ on $\overline{\left\langle
T\right\rangle}$ in terms of the parameters of $\varphi(z)$. We then prove
bounds for large and small $E$ numbers by lemma 2 and lemma 3. The proofs in
this section are algebraically lighter than the proof for a bound on
$\smash{\overline{\left\langle wT\right\rangle}}$, primarily due to two
reasons. The first is that the minimum principle on $T$, lemma 1, is not
required, and the second, the balance parameters do not have a $R$ dependence.
The lower bounds we prove on $\overline{\left\langle T\right\rangle}$,
including those already known, are illustrated in fig. 6.

$E$$\overline{\left\langle
T\right\rangle}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}E^{2/7}R^{-2/7}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}R^{-2/7}}$${\color[rgb]{0,0.498,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.498,0}R^{-1/3}}$${\color[rgb]{0,0.498,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.498,0}E^{-1}R^{-1}}$${\color[rgb]{0,0.44140625,0.73828125}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.44140625,0.73828125}R^{-1/4}\ln{R}^{-1/4}}$${\color[rgb]{0.84765625,0.3203125,0.09375}\definecolor[named]{pgfstrokecolor}{rgb}{0.84765625,0.3203125,0.09375}R^{-5/17}}$ABCD$E\leq
R^{-2/3}$$R^{-2/3}\leq E$$E_{1}$$E_{m}$$0$$0$$\frac{1}{12}$ Figure 6:
Illustration of the lower bounds on the mean temperature between no-slip and
isothermal boundary conditions at infinite $Pr$. The blue plot ( ) is the
bound derived in section 6.3, valid up to $E_{m}\sim 41.4487$ and $E_{1}=8$.
The bounds derived in section 6.4 are shown with a green plot ( ). The dashed
horizontal line denotes when all heat transport is by conduction and
$\overline{\left\langle T\right\rangle}=\frac{1}{12}$. In D ( ), we show the
previously known best bound for stress-free (top, red) (Whitehead & Doering,
2012) and no-slip (bottom, blue) (Whitehead & Doering, 2011a) at zero
rotation.

###  6.1 The auxiliary functional method

By the auxiliary functional method, we obtain an explicit variational problem
that we solve to obtain a bound on $\overline{\left\langle T\right\rangle}$.
The following derivation, albeit in the language of the classic background
field method approach, appears in previous papers (Whitehead & Doering, 2011a,
2012). Here, we only present an outline within the auxiliary functional
framework.

Starting with the quadratic auxiliary functional

$\mathcal{V}\\{T\\}=\left\langle-\frac{1}{2}|T|^{2}+2\varphi(z)T\right\rangle,$
(92)

where $\varphi(z)$ is the background temperature field subject to the boundary
conditions

$\varphi(0)=\varphi(1)=0.$ (93)

Then, provided $\mathcal{V}\\{T(t)\\}$ is bounded along solutions of (4), then
the time derivative of the long-time average of $\mathcal{V}\\{T\\}$ is zero,
such that we can write

$\overline{\left\langle T\right\rangle}=L-\left(L-\overline{\left\langle
T\right\rangle+\tfrac{\textrm{d}}{\textrm{d}t}\mathcal{V}\\{T\\}}\right)\geq
L,$ (94)

where the inequality comes from assuming that, $L-\overline{\left\langle
T\right\rangle+\tfrac{\textrm{d}}{\textrm{d}t}\mathcal{V}\\{T\\}}\leq
L-\left\langle
T\right\rangle-\tfrac{\textrm{d}}{\textrm{d}t}\mathcal{V}\\{T\\}\leq 0$.
Noting that we again bound the terms in the long-time integral by the
pointwise in time maximum over all periodic $\boldsymbol{u}$ and $T$, where
$\bnabla\cdot\boldsymbol{u}=0$, subject to the boundary conditions (6). Hence,
after substituting for the Lie derivative of $\mathcal{V}\\{T\\}$ by use of
(4c) and rearranging, we have, after appropriate manipulations, the following
variational problem

$\overline{\left\langle
T\right\rangle}\geq\inf_{L,\varphi(z)}\Big{\\{}L\;|\;\mathcal{S}\\{\boldsymbol{u},T\\}\geq
0\quad\forall(\boldsymbol{u},T)\in\mathcal{H}_{+}\Big{\\}},$ (95)

where

$\mathcal{S}\\{\boldsymbol{u},T\\}:=\left\langle|\bnabla
T|^{2}+\varphi^{\prime}(z)wT-\varphi^{\prime}(z)\partial_{z}T+\varphi(z)-L\right\rangle.$
(96)

In this case, the spectral constraint is given by

$\left\langle|\bnabla T|^{2}+2\varphi^{\prime}(z)wT\right\rangle\geq 0.$ (97)

Then, by optimising the linear terms in (96) the explicit expression for $L$
is given

$\displaystyle L:=$
$\displaystyle~{}2\left\langle\varphi(z)\right\rangle-\left\langle|\varphi^{\prime}(z)|^{2}\right\rangle.$
(98)

###  6.2 Preliminaries

To establish a lower bound on $\overline{\left\langle T\right\rangle}$, we
take the following class of background fields

$\varphi(z):=\begin{dcases}\tfrac{1}{2}(1-\delta)z,&0\leq z\leq\delta,\\\
\tfrac{1}{2}\delta(1-\delta),&\delta\leq z\leq 1-\delta,\\\
\tfrac{1}{2}(1-\delta)(1-z),&1-\delta\leq z\leq 1.\end{dcases}$ (99)

The background field is entirely determined by $\delta\in(0,\frac{1}{3})$, the
width of the boundary layers at the top and bottom of the domain, sketched in
fig. 7. In contrast to section 5, no advantage follows from background fields
with boundary layers of different widths.

$z$$\varphi(z)$$1$$\delta$$1-\delta$$\frac{1}{2}\delta(1-\delta)$ Figure 7:
Sketches of the functions $\varphi(z)$ in (99) used to prove (3), where
$\delta$ is the boundary layer widths.

The lower bounds on $\overline{\left\langle T\right\rangle}$ for different
regimes of the Ekman number rely on different estimates of the velocity in the
spectral constraint (97) given the diagnostic equations (7). Given the choice
of background profile (99), the expression for the lower bound $L=L(\delta)$
is unchanged based on the estimates we use to demonstrate the non-negativity
of the spectral constraint for different regimes of the Ekman number. So, we
now estimate $L$ in (98) given (99). Substituting for $\varphi(z)$ from (99),
we get

$L=\frac{1}{2}\delta(1-\delta)^{2}.$ (100)

Since $\delta\leq\frac{1}{3}$, then the estimate $1-\delta\geq\frac{2}{3}$,
gives

$L\geq\frac{2}{9}\delta\,.$ (101)

###  6.3 Large Ekman numbers

We can now proceed to establish a lower bound on $\overline{\left\langle
T\right\rangle}$ by establishing the conditions under which the spectral
constraint of (95) is satisfied, given the choice of background temperature
field (99) and by use of lemma 2. The estimates in lemma 2 apply to (7) in
Fourier space so we exploit horizontal periodicity and substitute (50) into
(97) such that the spectral constraint in Fourier space is

$\int^{1}_{0}|{T}^{\prime}_{\boldsymbol{k}}|^{2}+k^{2}|{T}_{\boldsymbol{k}}|^{2}+2\varphi^{\prime}(z)\textrm{Re}\\{{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}^{*}\\}\textrm{d}z\geq
0.$ (102)

Substituting for $\varphi(z)$ and using the fact that $\delta\leq\frac{1}{3}$
gives $-(1-\delta)\geq-1$, the sign-indefinite of (102) is

$\displaystyle
2\int^{1}_{0}\varphi^{\prime}(z)\textrm{Re}\\{{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}^{*}\\}\textrm{d}z$
$\displaystyle\geq-2\int^{1}_{0}|\varphi^{\prime}(z){w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z$
$\displaystyle\geq-\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z-\int^{1}_{1-\delta}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\,.$
(103)

Due to the symmetry in the boundary conditions and $\varphi(z)$, we only
demonstrate estimates on the integral at the lower boundary since that at the
top gives an identical contribution. Using the estimates of (53) and (54) on
$w_{\boldsymbol{k}}$ and $T_{\boldsymbol{k}}$, lemma 2 and Youngs’ inequality,
we get

$\displaystyle\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z$
$\displaystyle\leq\frac{1}{2}\int^{\delta}_{0}z^{5/2}\textrm{d}z\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}$
$\displaystyle=\frac{1}{14}\delta^{7/2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}+\frac{1}{14}\delta^{7/2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{w}^{\prime\prime}_{\boldsymbol{k}}\rVert_{\infty}$
$\displaystyle\leq\frac{\delta^{7/2}}{14}c_{1}R(1+\tfrac{1}{4}E^{-2})^{1/4}\lVert{T}_{\boldsymbol{k}}^{\prime}\rVert_{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}$
$\displaystyle\quad+\frac{\delta^{7/2}}{14}\lVert{T}_{\boldsymbol{k}}^{\prime}\rVert_{2}c_{2}\big{(}R\sqrt{k}\lVert{T}_{\boldsymbol{k}}\rVert_{2}+RE^{-1}\lVert{T}_{\boldsymbol{k}}\rVert_{2}\big{)}.$
(104)

Taking the term of order $\sqrt{k}$, estimating by use of Youngs’ inequality
twice and (57) gives

$\displaystyle\frac{1}{14}\delta^{7/2}R\sqrt{k}c_{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}$
$\displaystyle\leq\frac{\sqrt{2}}{2}k\lVert{T}_{\boldsymbol{k}}\rVert_{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}+\frac{c_{2}^{2}}{392\sqrt{2}}\delta^{7}R^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}$
$\displaystyle\leq\frac{1}{2}k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{1}{4}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{c_{2}^{2}}{784}\delta^{7}R^{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(105)

Therefore, the integral from 0 to $\delta$ simplifies to

$\displaystyle\int^{\delta}_{0}|{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}|\textrm{d}z\leq$
$\displaystyle~{}\frac{1}{2}k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{1}{4}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{c_{2}^{2}}{784}\delta^{7}R^{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}$
$\displaystyle+\frac{\sqrt{2}}{28}c_{1}\delta^{7/2}R(1+\tfrac{1}{4}E^{-2})^{1/4}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}+\frac{\sqrt{2}}{28}c_{2}\delta^{7/2}RE^{-1}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(106)

By equivalent estimates at the upper boundary, we get the same upper bound
such that (6.3) becomes

$\displaystyle
2\int^{1}_{0}\varphi^{\prime}(z)\textrm{Re}\\{{w}_{\boldsymbol{k}}{T}_{\boldsymbol{k}}^{*}\\}\textrm{d}z\geq$
$\displaystyle-k^{2}\lVert{T}_{\boldsymbol{k}}\rVert_{2}^{2}-\frac{1}{2}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}-\Bigg{(}\frac{c_{2}^{2}}{392}\delta^{7}R^{2}$
$\displaystyle+\frac{\sqrt{2}}{14}c_{1}\delta^{7/2}R(1+\tfrac{1}{4}E^{-2})^{1/4}+\frac{\sqrt{2}}{14}c_{2}\delta^{7/2}RE^{-1}\Bigg{)}\lVert{T}^{\prime}_{\boldsymbol{k}}\rVert_{2}^{2}.$
(107)

Finally, we use the estimate in (60), which places an upper limit on $E=E_{m}$
in (61), for which the bounds are valid. Then, substituting (107) into (102)
with (60), the spectral constraint becomes

$\frac{1}{2}-\frac{c_{2}^{2}}{392}\delta^{7}R^{2}-\frac{\sqrt{2}}{7}c_{2}\delta^{7/2}RE^{-1}\geq
0.$ (108)

In (108), two possible choices of $\delta=\delta(R,E)$ guarantee the non-
negativity of the left-hand side. If,

$\frac{c_{2}^{2}}{392}\delta^{7}R^{2}\leq\frac{\sqrt{2}}{7}c_{2}\delta^{7/2}RE^{-1},$

then from we (108) we have that

$\delta\leq\left(\frac{7\sqrt{2}}{8c_{2}}\right)^{2/7}R^{-2/7}E^{2/7}.$ (109)

Taking (109) as large as possible, substituting back into (108) and
rearranging, we find that the spectral constraint holds when $E\leq 8$. In the
opposite scenario where,

$\frac{\sqrt{2}}{7}c_{2}\delta^{7/2}RE^{-1}\leq\frac{c_{2}^{2}}{392}\delta^{7}R^{2},$

substituted back into (108) gives that

$\delta\leq\left(\frac{196}{c_{2}^{2}}\right)^{1/7}R^{-2/7},$ (110)

which holds for $E\geq 8$. In summary, we have that

$\delta=\begin{dcases}\left(\frac{7\sqrt{2}}{8c_{2}}\right)^{2/7}R^{-2/7}E^{2/7},&E\leq
8,\\\ \left(\frac{196}{c_{2}^{2}}\right)^{1/7}R^{-2/7},&E\geq 8,\end{dcases}$
(111)

which we can finally substitute in to (101) remembering that
$\overline{\left\langle T\right\rangle}\geq L$ to obtain that

$\overline{\left\langle
T\right\rangle}\geq\begin{dcases}d_{10}R^{-2/7}E^{2/7},&E\leq 8,\\\
d_{11}R^{-2/7},&8\leq E\leq E_{m},\end{dcases}$ (112)

where $d_{10}$ and $d_{11}$ are in appendix A. A final check is on the
validity of the bounds given choices made in section 6.2 that $\delta$ is in
$(0,\frac{1}{3})$. Given (111) the bound obtained in (112) holds for all
$R\geq 7.1363$ when $E\leq 8$ and $R\geq 10.0922$ when $E\geq 8$.

###  6.4 Small Ekman numbers

Next, we move on to the proof of the lower bound on $\overline{\left\langle
T\right\rangle}$ in (3) valid for a small $E$. In this regime, we will use
lemma 3, which does not require estimates in Fourier space. Starting with the
spectral constraint (97) and substituting for $\varphi(z)$ from (99), the
sign-indefinite term becomes

$2\left\langle\varphi^{\prime}(z)wT\right\rangle\geq-2\left\langle|\varphi^{\prime}(z)wT|\right\rangle\geq-\left\langle\int^{\delta}_{0}|wT|\textrm{d}z\right\rangle_{h}-\left\langle\int^{1}_{1-\delta}|wT|\textrm{d}z\right\rangle_{h}.$
(113)

In section 5.5, we established the estimate (79) and can directly substitute
into integral at the lower boundary in (113) to obtain

$\left\langle\int^{\delta}_{0}|wT|\textrm{d}z\right\rangle_{h}\leq\left[\frac{E\,R}{\sqrt{2}\pi}\left(\delta+\frac{2\delta^{3/2}}{3}\right)+\frac{1}{9\pi}\delta^{3}R\right]\left\langle|\bnabla
T|^{2}\right\rangle.$ (114)

The integral at the upper boundary gives an identical estimate such that the
spectral constraint of (97) becomes

$\displaystyle\left\langle|\bnabla T|^{2}+2\varphi^{\prime}(z)wT\right\rangle$
$\displaystyle\geq\left\langle|\bnabla
T|^{2}-2|\varphi^{\prime}(z)wT|\right\rangle$
$\displaystyle\geq\left\langle|\bnabla
T|^{2}\right\rangle-2\left[\frac{E\,R}{\sqrt{2}\pi}\left(\delta+\frac{2\delta^{3/2}}{3}\right)+\frac{1}{9\pi}\delta^{3}R\right]\left\langle|\bnabla
T|^{2}\right\rangle\geq 0.$ (115)

Therefore, the spectral constraint becomes the condition

$1-\frac{2E\,R}{\sqrt{2}\pi}\left(\delta+\frac{2\delta^{3/2}}{3}\right)-\frac{1}{9\pi}\delta^{3}R\geq
0.$ (116)

Given that $\delta\leq\frac{1}{3}$, we use the estimate that
$1+2\delta^{1/2}/3\leq\frac{\sqrt{2}\pi}{3}$ to get the condition,

$1-\frac{2}{3}ER\delta-\frac{1}{9\pi}\delta^{3}R\geq 0.$ (117)

Once more, the condition admits $\delta=\delta(R,E)$ valid for two different
regimes. If we take

$\frac{2}{3}ER\delta\leq\frac{1}{9\pi}\delta^{3}R,$

then (117) is non-negative if

$\delta\leq\left(\frac{9\pi}{2}\right)^{1/3}R^{-1/3},$ (118)

where when taking $\delta$ as large as possible the choice holds when
$E\leq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3}$. Whereas in the
case where

$\frac{1}{9\pi}\delta^{3}R\leq\frac{2}{3}ER\delta,$

then

$\delta\leq\frac{3}{4}R^{-1}E^{-1},$ (119)

for all $E\geq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3}$.
Therefore, the spectral constraint is satisfied if

$\delta=\begin{dcases}\left(\frac{9\pi}{8}\right)^{1/3}R^{-1/3},&E\leq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3},\\\
\frac{3}{4}R^{-1}E^{-1},&E\geq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3}.\end{dcases}$
(120)

Substituting into (101) and remembering further that $\overline{\left\langle
T\right\rangle}\geq L$, gives

$\overline{\left\langle
T\right\rangle}\geq\begin{dcases}d_{12}R^{-1/3},&E\leq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3},\\\
d_{13}R^{-1}E^{-1},&E\geq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3},\end{dcases}$
(121)

where $d_{12}$ and $d_{13}$ are in appendix A. Finally, for completeness, we
verify that the choice of $\delta\in(0,\frac{1}{3})$ made in section 6.2 is
not restrictive. Given (120) the bound obtained in (121) holds for all $R\geq
95.4259$ when $E\leq\frac{3}{4}\left(\frac{2}{9\pi}\right)^{1/3}R^{-2/3}$ and
for all $R\geq 1$ otherwise.

##  7 Discussion

###  7.1 Regimes of the bounds

Owing to the use of two different estimates (lemma 2 and lemma 3), we need to
consider how the bounds overlap. Furthermore, we can compare the heuristic
scaling laws of (30a) and (31a), with the lower bounds (74), (91), (112) and
(121). To this effect, first, we plot the regimes in which the bounds on
$\mathcal{F}_{B}$ and $\overline{\left\langle T\right\rangle}$ overlap in fig.
8. For ease of understanding, we split the space of $E$ and $R$ into four
regimes. Regime I, where $R$ and $E$ are large, corresponds to a slowly
rotating buoyancy-dominated flow. Regime II is the solid body rotation of the
fluid since $R$ and $E$ are small. Regime III, where $R$ is large and $E$
small, is rotation-dominated convection provided $R>R_{L}$, and the same
applies to regime IV, where $E$ is small and $R$ cannot get too large. Then,
we will compare the bounds to the heuristic scaling scales for rotation-
dominated convection due to internal heating.

(a)(b)

Figure 8: The Rayleigh and Ekman number regimes diagrams for the bounds (74),
(91), (112) and (121). In (a), we plot the regimes for the bounds on
$\mathcal{F}_{B}$ and in (b) the regimes for the bounds on
$\overline{\left\langle T\right\rangle}$. The blue solid lines ( ) corresponds
to $E=0.8201\,R^{-2}$ in (a) and $E=0.3102\,R^{-2/3}$ in (b) and the
horizontal red solid lines ( ) to $E_{0}=5.4927$ in (a) and $E_{1}=8$ in (b).
In both figures, the horizontal yellow lines ( ) are $E_{m}=41.4487$, the
vertical dashed lines $R_{L}=26926.6$, the green dashed lines ( ) the
asymptotic result of (27) and the green dot dashed line ( ) denotes the
expected line above which the flow is linearly unstable for no-slip
boundaries. The four main regimes are labelled I to IV.

In fig. 8(a), the blue solid line shows $E=0.8201\,R^{-2}$, the red line is
the constant $E_{0}=5.4927$, whereas in fig. 8(b) the blue line shows
$E=0.3102R^{-2/3}$, and the red line is the constant $E_{1}=8$. In both (a)
and (b), the yellow line is the constant $E_{m}=41.4487$ from (61), the dotted
vertical line, $R_{L}=26926.6$ and the dashed green line is the asymptotic
result of (27). The dot-dashed line is a sketch of the expected $R_{L}$ for
no-slip boundaries valid at each $R$, for all $R$ below the green curves,
rotation inhibits convection.

Starting with fig. 8(a), the bounds of (74) and (91) split the diagram into
four and we evaluate the best bound in each regime. For regime I, where
$E\gtrsim R^{-2}$ and $E\geq E_{0}$, the only valid bound is
$d_{4}R^{-2/3}+d_{5}R^{-1/2}|\ln(1-d_{6}R^{-1/3})|$, however, this bound holds
only up to $E\leq E_{m}$. The scaling of the slowly rotating convection bound
matches the zero rotation bound of Arslan & Rojas (2024). In regime II,
$E\lesssim R^{-2}$ and $E\geq E_{0}$, the best bound would match I, but the
regime is below $R_{L}$ when $E=\infty$, so no convection occurs, and
$\mathcal{F}_{B}=\frac{1}{2}$. In regime III, $E\gtrsim R^{-2}$ and $E\leq
E_{0}$, the only bound is
$d_{1}R^{-2/3}E^{2/3}+d_{2}R^{-1/2}E^{1/2}|\ln(1-d_{3}R^{-1/3}E^{1/3})|$,
provided $R>R_{L}$. Finally, in regime IV, where $E\lesssim R^{-2}$ and $E\leq
E_{0}$, the best bound is $d_{7}R^{-1}+d_{8}R^{-4/5}|\ln(1-d_{9}R^{-2/5})|$,
however for the entire region $R<R_{L}$ and no convection occurs.

Moving on to fig. 8(b), which represents the bounds (112) and (121). In regime
I, $E\gtrsim R^{-2/3}$ and $E\geq E_{1}$, there are two bounds of
$d_{13}R^{-1}E^{-1}$ and $d_{11}R^{-2/7}$, where from a comparison of the two
the better bound is $d_{11}R^{-2/7}$, due to the requirement of $E\gtrsim
R^{-2/3}$. In regime II, $E\lesssim R^{-2/3}$ and $E\geq E_{1}$, and the
dominant bound would be $d_{11}R^{-2/7}$, but the region is below the linear
stability limit, so $\overline{\left\langle T\right\rangle}=\tfrac{1}{12}$. In
regime III, $E\gtrsim R^{-2/3}$ and $E\leq E_{1}$, there are two bounds of
$d_{10}R^{-2/7}E^{2/7}$ and $d_{13}R^{-1}E^{-1}$, of which the second is
better for all $R>R_{L}$. Remember that the dashed green line scales as
$R^{-3/4}$, so, for sufficiently large $R$, the entire regime of III
corresponds to convecting flows. The asymptotic result (27) intersects the
blue line at approximately $R=8.569\times 10^{18}$ and $E=7.407\times
10^{-14}$, which would be higher $R$ and smaller $E$, for no-slip boundaries.
In regime IV, $E\lesssim R^{-2/3}$ and $E\leq E_{1}$, there are two bounds but
the better one is $d_{12}R^{-1/3}$. Note that unlike for the bounds on
$\mathcal{F}_{B}$, for $\overline{\left\langle T\right\rangle}$ in regime IV,
there exists a region where $R>R_{L}$, therefore for rotating convection,
there are two possible scaling laws, one in regime IV and other in III of
$d_{12}R^{-1/3}$ and $d_{13}R^{-1}E^{-1}$ respectively. Table 1 summarises the
above discussion.

Table 1: Summary of the lower bounds on $\mathcal{F}_{B}$ and $\overline{\left\langle T\right\rangle}$ with respect to the regimes in fig. 8. The constants in the bounds ((74),(91),(112) and (121)) are omitted for brevity, while $E_{0}=5.4927$ and $E_{1}=8$. Region I corresponds to buoyancy-dominated convection, II to no convection, III to buoyancy or rotation-dominated convection and IV to no or rotation-dominated convection. Regime |  Condition |  Bound on $\mathcal{F}_{B}$ |  Condition |  Bound on $\overline{\left\langle T\right\rangle}$
---|---|---|---|---
I |  $E\gtrsim R^{-2}$, $E\geq E_{0}$ |  $R^{-2/3}+R^{-1/2}|\ln(1-R^{-1/3})|$ |  $E\lesssim R^{-2/3}$, $E\geq E_{1}$ |  $R^{-2/7}$
II |  $E\lesssim R^{-2}$, $E\geq E_{0}$ |  $\frac{1}{2}$ |  $E\lesssim R^{-2/3}$, $E\geq E_{1}$ |  $\frac{1}{12}$
III |  $E\gtrsim R^{-2}$, $E\leq E_{0}$ |  $R^{-2/3}E^{2/3}+R^{-1/2}E^{1/2}|\ln(1-R^{-1/3}E^{1/3})|$ |  $E\gtrsim R^{-2/3}$, $E\leq E_{1}$ |  $R^{-1}E^{-1}$
IV |  $E\lesssim R^{-2}$, $E\leq E_{0}$ |  $\frac{1}{2}$ |  $E\lesssim R^{-2/3}$, $E\leq E_{1}$ |  $R^{-1/3}$

Finally, consider the heuristic scalings of $\overline{\left\langle
T\right\rangle}\sim R^{-3/5}E^{-4/5}$ from (30b) and $\mathcal{F}_{B}\sim
R^{-3/10}E^{-2/5}$ from (31b), that hold for rotation dominated convection. In
both scaling laws, if we fix $R$ and take $E\rightarrow 0$, we would obtain
the uniform upper bounds for both quantities. The lower bounds closest to the
heuristic scaling are those in region III (table 1). The bounds do not
contradict the heuristic arguments and are rigorous lower bounds for future
studies of rotating IHC. It is insightful to consider the small $E$ limit
where $E\sim R_{L}^{-3/4}$, and have

$\displaystyle\overline{\left\langle
T\right\rangle}\sim\left(\frac{R}{R_{L}}\right)^{-3/5},\qquad\text{and}$
(122a) $\displaystyle\mathcal{F}_{B}\sim\left(\frac{R}{R_{L}}\right)^{-3/10},$
(122b)

whereas the rigorous bounds in regime III (table 1) are at best

$\displaystyle\overline{\left\langle
T\right\rangle}\gtrsim\left(\frac{R}{R_{L}}\right)^{3/14}R^{-1/2},\qquad\text{and}$
(123a)
$\displaystyle\mathcal{F}_{B}\gtrsim\left(\frac{R}{R_{L}}\right)^{1/2}R^{-7/6}+\left(\frac{R}{R_{L}}\right)^{3/8}R^{-7/8}\left|\ln{\left(1-\left(\frac{R}{R_{L}}\right)^{1/4}R^{-7/12}\right)}\right|.$
(123b)

Since $R$ is always a multiple of $R_{L}$, for $R>R_{L}$, the bounds in (123)
are smaller than the heuristic scaling laws (122). If the lower bounds are not
sharp, then this would motivate the question of how to improve the bounds,
which we now discuss with concluding remarks.

###  7.2 Conclusions

In this work we prove lower bounds on the mean temperature
$\overline{\left\langle T\right\rangle}$, ((112) and (121)) and the mean heat
flux out of the bottom boundary $\mathcal{F}_{B}$ ((74) and (91)), for
rotating uniform internally heated convection (IHC) in the limit of infinite
Prandtl number. Using the fact that the momentum equation in Rayleigh-Bénard
(RBC) and IHC is identical, we adapt estimates from Yan (2004) and Constantin
et al. (1999) to prove the first Rayleigh and Ekman number dependent bounds on
$\overline{\left\langle T\right\rangle}$ and $\mathcal{F}_{B}$ in IHC. By
application of the auxiliary functional method, we prove bounds that apply to
different regimes of buoyancy to rotation-dominated flows, summarised in fig.
8 and table 1. In addition to rigorous bounds, we demonstrate that the
critical Rayleigh number for linear stability, $R_{L}$, asymptotically scales
with the Ekman number as $E^{-4/3}$ when the marginally stable states are
steady. Furthermore, given the relation between $E$ and the critical Rayleigh
number for convection, we use heuristic arguments to propose a family of
scaling laws for rotating IHC valid at arbitrary $Pr$.

In contrast to previous applications of the background field method, there are
several unique features in the proofs of bounds in this work. Firstly, the
background temperature fields have boundary layers of different widths for the
proofs on $\mathcal{F}_{B}$, but not for $\overline{\left\langle
T\right\rangle}$. In particular, when we use lemma 2 we find that
$\delta=\varepsilon^{2}$, while when using lemma 3,
$\delta=\varepsilon^{5/2}$. The relation from using lemma 2 (section 5.4)
matches the predictions of our heuristic arguments (section 4). However,
whether or not the background profiles are optimal remains unknown and should
be addressed with numerical optimisation (Fantuzzi & Wynn, 2016; Fantuzzi,
2018; Fantuzzi et al., 2022). The bound obtained for $\mathcal{F}_{B}$ in the
slowly rotating regime (see regime I in fig. 8 and table 1) matches the best
known bound for zero rotation, while this is not the case for the bound on
$\overline{\left\langle T\right\rangle}$, but, both bounds only hold up to
$E\leq E_{m}=41.4487$. It would be interesting to investigate if a bound can
be proven that holds for all $E$ and matches bounds without rotation on
$\overline{\left\langle T\right\rangle}$ (Whitehead & Doering, 2011a) and
$\mathcal{F}_{B}$ (Arslan & Rojas, 2024) when $E\rightarrow\infty$.
Additionally, from table 1, we realise that using lemma 3 for a bound on
$\mathcal{F}_{B}$ is not as good as for $\overline{\left\langle
T\right\rangle}$ because of the requirement that $R>R_{L}\sim E^{-4/3}$. It
remains to be seen if one can find a bound on $\mathcal{F}_{B}$ for $E\leq
R^{-n}$ where $n<3/4$. The two lemmas give different bounds because they
estimate the second derivative of the vertical velocity in different $L^{p}$
norms and while lemma 2 is a pointwise estimate in $z$, lemma 3 is an integral
estimate over the whole domain.

While rigorous demonstrations of the validity of the results at arbitrary $Pr$
are not provided here, previous work in Tilgner (2022) outlines a strategy for
extending bounds from $Pr=\infty$ to finite $Pr$ for RBC. The author achieves
this under specific restrictions on $E$ and a numerical approximation of a
Greens’ function, making the bound semi-analytic at best. A similar approach
appears in Wang & Whitehead (2013) to extend bounds on RBC for stress-free
boundaries from infinite $Pr$ to arbitrary $Pr$ in 3D. The barrier to adapting
to IHC is the lack of a maximum principle on the temperature of the form
$\lVert T\rVert_{\infty}\leq c$, where $c=1$ for RBC. A proof of any maximum
principle for IHC is not known. Therefore, at best, we can conjecture that
akin to RBC, to highest order, the bounds in this work should hold for
arbitrary $Pr$.

In considering turbulent convection subject to rotation, a question of
interest is the behaviour in the limit of rapidly rotating convection. Rapidly
rotating IHC could be investigated by taking the approach of the non-
hydrostatic quasigeostrophic approximation (Julien et al., 1996; Sprague et
al., 2006; Julien et al., 2016) to (4). It is worth noting that the bounds for
the rapidly rotating limit in RBC apply to arbitrary $Pr$, making the results
relevant for geophysical flows. However, for geophysical applications, in
addition to rapid rotation, IHC in a spherical geometry is of importance. No
rigorous work on the turbulent state of such a system has been carried out,
and the change in the bounds with a variation in the geometry is an exciting
avenue for future research.

For any result obtained with a bounding method, an important question is on
the sharpness of the bounds. One way to gain insight into the sharpness of the
bounds (2) and (3) is with numerical simulations. A numerical study of the
parameter space would provide valuable insight into the nature of heat
transport in uniform rotating IHC, especially given the lack of data, both
numerical and experimental, on such a flow. Then, proof of better bounds, by
moving away from quadratic auxiliary functionals, and hence the background
method, would also answer the question of sharpness. In general, mathematical
improvements are obtained in one of two ways: either by changes to the
variational problem, such that the expressions for the bounds and the spectral
constraint change, or by novel estimates of the flow quantities. The latter
method is more mathematically challenging, but the first can be achieved with
new physical insights. For example, given that additional constraints, like
minimum and maximum principles, improve bounds for convection (Otto & Seis,
2011; Arslan et al., 2021b), it would be interesting to see if information
about rotating flows can be further leveraged to construct a variational
problem from (4), that yields better bounds. Beyond the relation between the
$R_{L}$ and $E$, a trait of rotating flows is the Taylor-Proudman theorem,
which could form the basis of an additional constraint which improves the
bounds. Also, alternative auxiliary functionals might bear fruit in studying
bounds on rotating convection. More concretely, the linear stability analysis
reveals the importance of the vertical vorticity, and functionals that
incorporate vorticity may provide insight into improved bounds, especially
given the conjecture in Chernyshenko (2023) for the use of helicity in the
auxiliary functional. Similar functionals appear in studies on nonlinear
stability of rotating RBC (Galdi & Straughan, 1985; Giacobbe & Mulone, 2014).

As a final remark, recent work has highlighted novel results when the internal
heating is non-uniform (Lepot et al., 2018; Bouillaut et al., 2022; Song et
al., 2022; Arslan et al., 2024). The change in the physics or bounds due to
distributed heating or cooling would be an interesting future line of
research. From the perspective of the PDE, extending the setup in (4c) to
arbitrary heating profiles would be the natural next step when studying bounds
on the long-time behaviour of $\overline{\left\langle T\right\rangle}$ and
$\smash{\overline{\left\langle wT\right\rangle}}$.

[Acknowledgments] The author thanks Andrew Jackson for fruitful discussions,
insights and for commenting on the manuscript and Fabian Burnman for reading
and commenting on the manuscript.

[Funding]The author acknowledges funding from the European Research Council
(agreement no. 833848-UEMHP) under the Horizon 2020 program and the Swiss
National Science Foundation (grant number 219247) under the MINT 2023 call.

[Declaration of interests] The author reports no conflict of interest.

##  Appendix A Table of constants

For clarity in the proofs of the lower bounds on $\mathcal{F}_{B}$ in (2) and
$\overline{\left\langle T\right\rangle}$ in (3), here we collate the constants
that appear in the bounds. References to the precise equations where they
appear are included.

Bound | Eq. number | Constant | Value
---|---|---|---
$\mathcal{F}_{B}$ | (74) | $d_{1}$ | $\frac{18-7\sqrt{3}}{36}\left(\frac{7\sqrt{6}}{16c_{2}}\right)^{2/3}$
| (74) | $d_{2}$ | $\frac{1}{12}\left(\frac{1}{6}\right)^{1/12}\sqrt{\frac{21}{c_{2}}}$
| (74) | $d_{3}$ | $\left(\frac{7\sqrt{6}}{12c_{2}}\right)^{1/3}$
| (74) | $d_{4}$ | $\frac{18-7\sqrt{3}}{36}\left(\frac{49\sqrt{2}}{2c_{2}^{2}}\right)^{1/3}$
| (74) | $d_{5}$ | $\frac{\sqrt{7}}{3}\Big{(}\frac{9}{32\sqrt{2}c_{2}^{5}}\Big{)}^{1/12}$
| (74) | $d_{6}$ | $\left(\frac{392\sqrt{2}}{9c_{2}^{2}}\right)^{1/6}$
| (91) | $d_{7}$ | $\frac{9\pi(18-7\sqrt{3})}{288}$
| (91) | $d_{8}$ | $\frac{3\pi}{8}\left(\frac{8}{9\pi}\right)^{1/5}$
| (91) | $d_{9}$ | $\left(\frac{9\pi}{8}\right)^{2/5}$
$\overline{\left\langle T\right\rangle}$ | (112) | $d_{10}$ | $\frac{2}{9}\left(\frac{7\sqrt{2}}{8c_{2}}\right)^{2/7}$
| (112) | $d_{11}$ | $\frac{2}{9}\left(\frac{196}{c_{2}^{2}}\right)^{1/7}$
| (121) | $d_{12}$ | $\frac{2}{9}\left(\frac{9\pi}{8}\right)^{1/3}$
| (121) | $d_{13}$ | $\frac{1}{6}$

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