2.1. Configuration and governing equations

Figure 1.

(a) A schematic diagram of rotating IHC between two isothermal parallel plates with stress-free boundary conditions. The uniform heating source is denoted by $S$ , and $H$ is the separation between the two plates in the direction of gravity, $g$ . Horizontal periodicities are $L_x H$ and $L_y H$ . The convection is subject to global rotation $\varOmega$ . Without loss of generality, the boundary temperature is set to $T_b = 0$ . Circles indicate flow structures with a characteristic vertical scale $H$ and horizontal scale $E^{1/3}H$ . (b) The same flow structures shown in dimensionless coordinates. The vertical and horizontal length units are $H$ and $E^{1/3}H$ , respectively, with $E\ll 1$ .

As illustrated in figure 1(a), we consider an incompressible fluid between two parallel plates separated by a distance $H$ with horizontal periodicities $L_x H$ and $L_y H$ . The fluid is characterised by reference density $\rho$ , thermal diffusivity $\kappa$ , kinematic viscosity $\nu$ , specific heat $c_p$ , and thermal expansivity $\alpha$ . Convection is driven by a uniform volumetric heat source $S$ under a constant gravitational acceleration $-g \hat {\boldsymbol{z}}$ . The system rapidly rotates about the vertical axis with constant angular velocity $\varOmega$ .

To describe rapid rotation, we use an asymptotically reduced set of equations to model the convection of a Boussinesq fluid in the limit of infinite Prandtl number $\textit{Pr}$ . The Prandtl number is defined in (2.1) and is the ratio of viscous to thermal diffusivity. The other control parameters are the Rayleigh number $R$ and Ekman number $E$ , where $R$ quantifies the destabilising effect of heating to the stabilising effect of diffusion, and $E$ gives the effect of inertia compared to rotation. These control parameters are defined as

(2.1) \begin{equation} R = \frac {\alpha g S H^5}{\rho c_p \nu \kappa ^2}, \quad E = \frac {\nu }{2\varOmega H^2}, \quad \textit{Pr}= \frac {\nu }{\kappa }. \end{equation}

An additional key control parameter characterising the dynamics is the reduced Rayleigh number (Julien et al. Reference Julien, Knobloch and Werne1998) measuring the degree of supercriticality:

(2.2) \begin{equation} \mathcal{R} = R E^{4/3}. \end{equation}

To non-dimensionalise, we choose the vertical length scale to be $H$ , and the horizontal length scale to be $E^{1/3} H$ . The effect of this anisotropic non-dimensionalisation on rotation-constrained columnar flow structures is illustrated in figure 1(b). The time and temperature scales are $E^{2/3} H^2 / \kappa$ and $S H^2/\rho c_p \kappa$ , respectively. Then in the asymptotic limit of rapid rotation, the equations for convection at infinite $\textit{Pr}$ (see Appendix A for a derivation) are

(2.3a) \begin{equation} \partial _Z\psi ={\mathcal{R}}\theta +{\nabla} _h^2w, \end{equation}

(2.3b) \begin{equation} -\partial _Zw={\nabla} _h^2\zeta , \end{equation}

(2.3c) \begin{equation} \zeta ={\nabla} _h^2\psi , \end{equation}

(2.3d) \begin{equation} \partial _t\theta + J\left [\psi ,\theta \right ]+w\partial _Z \varTheta ={\nabla} _h^2\theta , \end{equation}

(2.3e) \begin{equation} E^{-{2}/{3}}\,\partial _t \varTheta +\partial _Z(\overline {w\theta })=\partial _Z^2 \varTheta +1. \end{equation}

Here, $J[\psi ,{\cdot }\,]:= \partial _x\psi \,\partial _y({\cdot }) - \partial _y\psi \,\partial _x({\cdot })$ , $\boldsymbol{\nabla} _h({\cdot }):= \hat {\boldsymbol{x}}\,\partial _x({\cdot })+\hat {\boldsymbol{y}}\,\partial _y({\cdot })$ and ${\nabla} _h^2 := \partial _x^2 + \partial _y^2$ denotes the horizontal Laplacian. Then $\psi (x,y,Z,t)$ is the horizontal streamfunction, with the horizontal velocity being $-\boldsymbol{\nabla} _h\times (\psi \hat {\boldsymbol{Z}})$ . Here, $w(x,y,Z,t)$ is the vertical velocity, $\zeta (x,y,Z,t)$ is the vertical vorticity, $\varTheta (Z,t)$ is the horizontally averaged temperature field, and $E^{1/3}\theta (x,y,Z,t)$ is the temperature anomaly relative to the horizontal average. It is noted that $w$ , $\zeta$ , $E^{1/3}\theta$ and $\varTheta$ represent the leading-order non-zero components of the corresponding fields in an asymptotic expansion in powers of $E^{1/3}$ , with the incompressibility condition satisfied by advection on higher-orders. If these variables are instead interpreted as the full fields, then (2.3a )–(2.3e ) incur an error of $O(E^{1/3})$ . The total temperature field $T(x,y,Z,t)$ is approximated by

(2.4) \begin{align} T=\varTheta +E^{{1}/{3}}\theta . \end{align}

As demonstrated in Appendix A, the horizontal means of $w$ , $\zeta$ and $\psi$ vanish, while $\theta$ has zero horizontal mean by definition. Hence $\overline {\theta } = \overline {w} = \overline {\zeta } = \overline {\psi } = 0$ .

In the derivation of (2.3), the geostrophic balance is assumed to constitute the leading-order force balance throughout the domain, thereby excluding any boundary-layer effects (see Appendix A). For the physical consistency of the resulting PDEs, we impose stress-free mechanical boundary conditions. For the temperature, the isothermal boundary conditions $T=T_b$ are applied. Without loss of generality, $T_b$ is set to zero. The boundary conditions are thus given by

(2.5) \begin{equation} w=\partial _Z\zeta =\partial _Z\psi =\theta =\varTheta = 0\quad \mathrm{at}\ Z=0\ \mathrm{and}\ Z=1. \end{equation}

2.2. Main theorems

Having defined the governing equations and boundary conditions, here we state the two theorems proved in this work.

Theorem 1 (Lower bound on mean temperature). On the domain $V=[0,E^{-1/3}L_x]\times [0,E^{-1/3}L_y]\times [0,1]$ , let $\omega,\psi,\theta : V\times [0,\infty ) \to \mathbb{R}$ and $\varTheta:[0,1]\times [0,\infty ) \to \mathbb{R}$ satisfy the model defined in ( 2.3 ), with periodic boundary conditions in $x$ and $y$ , and subject to the boundary conditions ( 2.5 ). Let $\langle \varTheta \rangle _{V,t}$ be the volume and long-time average of $\varTheta$ as defined by ( 1.1c ). Then

(2.6) \begin{align} \langle \varTheta \rangle _{V,t}\geq \begin{cases} 1.13E^{-8/7}R^{-6/7}-1.61 E^{-12/7}R^{-9/7},&5.82 E^{-4/3}\leq R\lt 19.2E^{-4/3},\\[5pt] 0.66E^{-1}R^{-3/4}- 1.51E^{-2}R^{-3/2},& R\geq 19.2 E^{-4/3}, \end{cases} \end{align}

where $5.82 E^{-4/3}$ corresponds to a lower threshold for nonlinear instability of the flow. For $R\lt 5.82 E^{-4/3}$ , the conductive state is globally asymptotically stable, and $\langle \varTheta \rangle _{V,t}=1/12$ .

It is straightforward to verify that the lower bounds are positive for all $R$ over which the bounds hold.

Theorem 2 (Upper bound on mean heat flux). On the domain $V=[0,E^{-1/3}L_x]\times [0,E^{-1/3}L_y]\times [0,1]$ , let $\omega,\psi,\theta : V\times [0,\infty ) \to \mathbb{R}$ and $\varTheta:[0,1]\times [0,\infty ) \to \mathbb{R}$ satisfy the model defined in ( 2.3 ), with periodic boundary conditions in $x$ and $y$ , and subject to the boundary conditions ( 2.5 ). Let $\langle w\theta \rangle _{V,t}$ be the volume and long-time average of $w\theta$ as defined by ( 1.1c ). Then

(2.7) \begin{align} \langle {w\theta }\rangle _{V,t}\leq \begin{cases} \begin{aligned} 0.01 E^{8/7}R^{6/7}-0.25 +0.57 E^{-4/7}R^{-3/7}-0.37E^{-8/7}R^{-6/7} , &\\[7pt] 3.50E^{-4/3}\leq R\lt 30.2E^{-4/3},&\\[7pt] 0.04E^{4/5}R^{3/5}-0.25- 0.15 E^{-4/5}R^{-3/5}+3.95E^{-8/5}R^{-6/5}, &\\[7pt] R\geq 30.2 E^{-4/3}.& \end{aligned} \end{cases} \end{align}

For $R\lt 3.50E^{-4/3}$ , the conductive state is globally asymptotically stable, and $\langle w\theta \rangle _{V,t}=0$ .

2.3. Quantities of interest

In this subsection, we demonstrate the relevance of the quantities for which we have stated bounds in Theorem1 and Theorem2. The mean temperature $\langle \varTheta \rangle _{V,t}$ quantifies the thermal dissipation and effective mixing of the fluid, while the mean convective heat flux $\langle {w\theta }\rangle _{V,t}$ quantifies the asymmetry in heat flux out of the domain due to convection. To see this, we multiply (2.3d ) and (2.3e ) by $\theta$ and $\varTheta$ , respectively, apply the spatial average (1.1b ), integrate by parts, and sum to obtain

(2.8) \begin{equation} \frac {E^{-{2}/{3}}}{2} \frac {\mathrm{d}}{\mathrm{d} t}\langle \varTheta ^2+E^{{2}/{3}}\theta ^2\rangle _{V}=-\langle \left |\boldsymbol{\nabla} _h \theta \right |^2+(\partial _Z \varTheta )^2\rangle _{V}+\langle \varTheta \rangle _{V}, \end{equation}

where the left-hand side represents the rate of change of the thermal energy density, while the first term on the right-hand side corresponds to the thermal dissipation rate. Further applying the long-time average yields the mean energy equilibrium

(2.9) \begin{equation} \langle \left |\boldsymbol{\nabla} _h \theta \right |^2+(\partial _Z \varTheta )^2\rangle _{V,t} = \langle \varTheta \rangle _{V,t}. \end{equation}

On the other hand, multiplying (2.3a ) and (2.3b ) by $w$ and $\psi$ , respectively, and proceeding similarly leads to the identity

(2.10) \begin{equation} \langle \zeta ^2+\left |\boldsymbol{\nabla} _h w\right |^2\rangle _{V,t} = {\mathcal{R}}\langle w\theta \rangle _{V,t}, \end{equation}

with the left-hand side representing the dissipation arising from velocity gradients. As can be seen in (2.9) and (2.10), the mean temperature and convective heat transport are non-negative, and this gives uniform lower bounds $\langle w \theta \rangle _{V,t} \geq 0$ and $ \langle \varTheta \rangle _{V,t} \geq 0$ . For a uniform upper bound on $\langle w \theta \rangle _{V,t}$ , we multiply (2.3e ) by $Z$ , and take the average (1.1c ). With $\mathcal{F}_T$ and $\mathcal{F}_B$ defined as the heat fluxes exiting through the top and bottom boundaries, this yields

(2.11a) \begin{equation} \mathcal{F}_T:=\left . -\limsup _{\tau \rightarrow \infty }\frac {1}{\tau }\int ^{\tau }_0 \partial _Z \varTheta \right |_{Z=1} \mathrm{d}t =\frac {1}{2}+\langle w\theta \rangle _{V,t}, \end{equation}

(2.11b) \begin{equation} \mathcal{F}_B:= \left . \limsup _{\tau \rightarrow \infty }\frac {1}{\tau }\int ^{\tau }_0 \partial _Z \varTheta \right |_{Z=0} \mathrm{d}t =\frac {1}{2}-\langle w\theta \rangle _{V,t}. \end{equation}

Given that the temperature within the domain is pointwise non-negative, $T \geq T_b=0$ with any deviation decaying exponentially in time (Arslan et al. Reference Arslan, Fantuzzi, Craske and Wynn2021), the heat fluxes are also signed, thus we infer that $\langle w\theta \rangle _{V,t} \leq 1/2$ . The uniform upper bound on the mean temperature is given by the conductive solution of the system (2.3) subject to (2.5), where the conductive temperature profile is

(2.12) \begin{equation} \varTheta _c=\frac {1}{2}Z(1-Z). \end{equation}

It follows that $\langle \varTheta \rangle _{V,t} \leq \langle \varTheta _c \rangle _{V}=1/12$ , with a proof analogous to that in Goluskin & Spiegel (Reference Goluskin and Spiegel2012).

2.4. Horizontal Fourier domain

Before presenting the framework for deriving rigorous bounds on $\langle {\varTheta }\rangle _{V,t}$ and $\langle {w\theta }\rangle _{V,t}$ , we introduce a Fourier expansion in the horizontal direction. Let $\mathcal{K}$ denote the set of all non-zero horizontal wavevectors $\boldsymbol{k}=k_x\hat {\boldsymbol{x}}+k_y\hat {\boldsymbol{y}}$ permitted by the dimensionless periods $E^{-1/3}L_x$ and $E^{-1/3}L_y$ . For any scalar field denoted generically by $f$ , we write

(2.13a) \begin{align} f=\overline {f}+\sum _{\boldsymbol{k}\in \mathcal{K}}{f}_{\boldsymbol{k}}(Z,\,t)\,\mathrm{e}^{\mathrm{i} \left (k_xx+k_yy\right )}, \end{align}

where

(2.13b) \begin{align} \mathcal{K} = \left \{ (k_x,k_y) = E^{1/3}\left (\frac {2\pi m}{L_x}, \frac {2\pi n}{L_y}\right )\Bigg | (m,n) \in \mathbb{Z}^2 \,\& \, (m,n)\neq (0,0) \right \}\!. \end{align}

Therein, ${f}_{\boldsymbol{k}}$ represent the Fourier coefficients. Since $f$ is real-valued, the coefficients satisfy ${f}_{\boldsymbol{k}}^{*}={f}_{-\boldsymbol{k}}$ , with $({\cdot })^{*}$ denoting complex conjugation. Under the expansion (2.13b ), the combination of (2.3a )–(2.3c ) yields a diagnostic relation between $w_{\boldsymbol{k}}$ and $\theta _{\boldsymbol{k}}$ ,

(2.14) \begin{equation} \theta _{\boldsymbol{k}}=\frac {1}{{\mathcal{R}}}\left (k^2 {w}_{\boldsymbol{k}}-\frac {1}{k^4}\,\partial _Z^2 {w}_{\boldsymbol{k}}\right )\!, \end{equation}

in which $k=\sqrt {k_x^2+k_y^2}$ is the magnitude of each wavevector.

3.1. Bounding framework

To bound $\langle \varTheta \rangle _{V,t}$ , we follow the auxiliary functional method (Fantuzzi et al. Reference Fantuzzi, Arslan and Wynn2022), which is based on the observation that for any bounded functional $\mathcal{V}\{\varTheta ,\theta \}$ along the solutions of the governing equations (2.3), the time derivative of $\mathcal{V}\{\varTheta ,\theta \}$ has zero long-time average. Moreover, the time average of a quantity is bounded from below by its pointwise infimum, so that where $\mathcal{V}$ is independent of spatial coordinates,

(3.1) \begin{equation} \langle \varTheta \rangle _{V,t}=\limsup _{\tau \rightarrow \infty }\frac {1}{\tau }\int ^{\tau }_0 \langle \varTheta \rangle _{V}-\frac {\mathrm{d}{\mathcal{V}}}{\mathrm{d}t}\,\mathrm{d}t\geq \inf \left (\langle \varTheta \rangle _{V}-\frac {\mathrm{d}{\mathcal{V}}}{\mathrm{d}t}\right )\geq B_{1}. \end{equation}

Therefore, $B_{1}$ , which may ultimately be shown to be a function of $R$ and $E$ , serves as a lower bound for $\langle \varTheta \rangle _{V,t}$ , provided that there exists an auxiliary functional $\mathcal{V}\{\varTheta ,\theta \}$ such that

(3.2) \begin{equation} Q:= \langle \varTheta \rangle _{V}-\frac {\mathrm{d}\mathcal{V}}{\mathrm{d}t}-B_{1}\geq 0. \end{equation}

We consider the quadratic auxiliary functional

(3.3) \begin{equation} \mathcal{V}\{\varTheta ,\theta \} = \frac {\beta }{2} \big \langle \theta ^2+E^{-{2}/{3}}\varTheta ^2 \big \rangle _{V}-E^{-{2}/{3}}\left \langle \phi \,\varTheta \right \rangle _{V}, \end{equation}

where $\beta \gt 0$ is referred to as the balance parameter, and we call $\phi =\phi (Z)$ the background field, although in most prior work, $\phi /\beta$ is typically referred to as the background field. The boundedness of $\mathcal{V}\{\varTheta ,\theta \}$ along solutions of (2.3) is demonstrated in Appendix B. Here, $\phi$ is piecewise differentiable with a square-integrable derivative, and satisfies the homogeneous boundary conditions

(3.4) \begin{equation} \phi (0)=0,\quad \phi (1)=0. \end{equation}

Now, we show that condition (3.2) is enforced by a $B_{1}$ value given by an appropriate choice of $\phi$ , and that the resulting $B_1$ is a valid lower bound on $\langle \varTheta \rangle _{V,t}$ , provided that an associated quadratic form is non-negative.

Proposition 1 (Bound on $\langle \varTheta \rangle _{V,t}$ ). For a given background field $\phi :[0,1]\rightarrow \mathbb{R}$ satisfying the boundary conditions ( 3.4 ), the scalar $B_{1}$ provides a lower bound on the mean temperature, where

(3.5) \begin{equation} B_{1}:=\int _0^1 \phi -\frac {1}{4}\phi '^2\,\mathrm{d}Z, \end{equation}

provided that the so-called spectral constraint is also satisfied:

(3.6) \begin{equation} Q_{\boldsymbol{k}}\left \{w,\phi \right \}\geq 0,\quad \forall k\gt 0,\quad \forall \! \left .w \,\right | \,{w}(0)={w}(1)=0, \end{equation}

in which

(3.7) \begin{equation} Q_{\boldsymbol{k}}:= k^6\,\|{w}_{\boldsymbol{k}}\|^2_2+2\,\|{w}^{\prime }_{\boldsymbol{k}}\|^2_2+\frac {1}{k^6}\|{w}^{\prime \prime }_{\boldsymbol{k}}\|^2_2+{\mathcal{R}}\int _0^1\phi '\left (k^2\left |w_{\boldsymbol{k}}\right |^2-\frac {1}{k^4}\,{\rm Re}\{{w}^{\prime \prime }_{\boldsymbol{k}}{w}^*_{\boldsymbol{k}}\}\right ) \mathrm{d}Z. \end{equation}

Proof. From (2.3d )–(2.3e ), we start by calculating the Lie derivative of $\mathcal{V}$ in (3.3). By integration by parts, use of the boundary conditions (2.5) and fixing $\beta =1$ , we get that

(3.8) \begin{equation} \frac {\mathrm{d}\mathcal{V}}{\mathrm{d}t} = -\big \langle |\boldsymbol{\nabla} _h \theta |^2 + | \varTheta '|^2 + \phi ' w\theta + (Z- \phi ') \varTheta ' + \phi \big \rangle _{V}. \end{equation}

We proceed by substituting (3.8) into the definition of $Q$ in (3.2), and applying the horizontal Fourier expansion (2.13b ) to $w$ and $\theta$ . This yields the decomposition

(3.9) \begin{equation} Q\{\varTheta ,w,\theta \}=Q_0\{\varTheta \}+\sum _{\boldsymbol{k}\in \mathcal{K}}Q_{\boldsymbol{k}}\{w_{\boldsymbol{k}},\theta _{\boldsymbol{k}}\}, \end{equation}

where

(3.10a) \begin{align} Q_0=-B_{1}+\int _0^1 (|\varTheta '|^2-\phi ' \varTheta ' +\phi) \,\mathrm{d}Z, \end{align}

(3.10b) \begin{align} Q_{\boldsymbol{k}}= k^2\,\| {\theta }_{\boldsymbol{k}}\|^2_2+\int _0^1 \phi ' \,{\rm Re}\{ {w}_{\boldsymbol{k}} {\theta }_{\boldsymbol{k}}^*\}\, \mathrm{d}Z. \end{align}

Condition (3.2) is satisfied by requiring $Q_0 \geq 0$ and $Q_{\boldsymbol{k}} \geq 0$ independently. Noting that $Q_0$ is convex in $\varTheta '$ , we solve the associated Euler–Lagrange equation with respect to $\varTheta$ , subject to Dirichlet boundary conditions. The non-negativity condition on $Q_0$ is satisfied in the worst case by

(3.11) \begin{equation} B_{1} = \inf _{ \substack {\varTheta (Z) \\ \varTheta (0)= \varTheta (1)=0 } } \int _0^1(|\varTheta ' |^2-\phi '\varTheta ' +\phi) \,\mathrm{d}Z, \end{equation}

with the minimiser given by

(3.12) \begin{equation} \varTheta '_{\textit{opt}}=\frac {\phi '}{2}. \end{equation}

Then substituting (3.12) into (3.11) gives the form (3.5).

Next, in (3.10b ), we eliminate the spectral coefficient $\theta _{\boldsymbol{k}}$ using the diagnostic relation (2.14), and rearrange through integration by parts to obtain

(3.13) \begin{align} \begin{aligned} Q_{\boldsymbol{k}}={}&\frac {1}{{\mathcal{R}}^2}\left (k^6\,\|{w_{\boldsymbol{k}}}\|^2_2+2\,\|{w_{\boldsymbol{k}}^{\prime }}\|^2_2+\frac {1}{k^6}\|{w_{\boldsymbol{k}}^{\prime \prime }}\|^2_2\right ) \\ &{}+\frac {1}{{\mathcal{R}}}\int _0^1\phi '\left (k^2\left |w_{\boldsymbol{k}}\right |^2-\frac {1}{k^4}\,{\rm Re}\{{w_{\boldsymbol{k}}^{\prime \prime }}{w_{\boldsymbol{k}}}^*\}\right ) \mathrm{d}Z. \end{aligned} \end{align}

The condition $Q_{\boldsymbol{k}} \geq 0$ for all $\boldsymbol{k}$ is satisfied by the spectral constraint.

Maximising (3.5) over all choices of $\phi$ subject to the spectral constraint yields the largest lower bound attainable within our framework. This optimisation problem is convex, and while analytically intractable, can be numerically solved. To obtain explicit bounds with rigorous parametrisation, here we focus on proving Theorem1 by constructing a suboptimal ansatz for $\phi$ , but present results of the numerical optimisation, where no specific form of $\phi$ is prescribed, in § 5.

If $L_x$ and $L_y$ were prescribed, then it would only be necessary for $Q_{\boldsymbol{k}}\geq 0$ to hold for the corresponding subset of positive wavenumbers, in order for $B_{1}$ to be a valid bound. Here, we require $Q_{\boldsymbol{k}}\geq 0$ for every positive $k$ . The resulting bounds are thereby valid for arbitrary horizontal periods, and for flow structures with arbitrary horizontal length scales, provided that (2.3) remains representative for the dynamics.

3.2. Ansatz

We restrict $\phi (Z)$ to take the form

(3.14) \begin{equation} \phi := \begin{cases} ({A}/{\delta ^2})Z\left (2\delta -Z\right ), &0\leq Z\leq \delta ,\\[5pt] A,&\delta \lt Z\lt 1-\delta ,\\[5pt] ({A}/{\delta ^2})\left (1-Z\right )\left (2\delta +Z-1\right ), &1-\delta \leq Z\leq 1. \end{cases} \end{equation}

Here, $\phi (Z)$ is characterised by boundary layers of thickness $\delta \in (0, 1/2]$ at the top and bottom of the layer, and by the amplitude parameter $A\gt 0$ specifying the bulk value of $\phi$ (figure 2).

The choice of a background field with quadratic boundary layers and a constant interior is motivated by the following considerations. First, it renders the sign-indefinite integral in (3.7) identically zero in the bulk. Second, as will be demonstrated in § 3.4, the maximal lower bound attainable with (3.14) is exactly $1/12$ when $\mathcal{R}$ is below a certain threshold. This value is achieved with the choice $A=1/4$ and $\delta = 1/2$ , for which the optimiser $\varTheta _{\textit{opt}}$ in (3.12) is exactly equivalent to the conductive state (2.12), and the spectral constraint becomes equivalent to a sufficient condition for nonlinear stability based on the energy method. This is only possible by using the quadratic boundary layers in $Z$ .

Figure 2.

A sketch of the background profile (3.14) featuring quadratic boundary layers.

Substituting (3.14) into (3.5) and (3.7) leads to

(3.15) \begin{equation} B_{1}=A\left (1-\frac {2\delta }{3}\right )-\frac {2A^2}{3\delta }, \end{equation}

along with

(3.16) \begin{equation} Q_{\boldsymbol{k}}= \underbrace { k^6\,\|{w}_{\boldsymbol{k}}\|_2^2+2\,\|{w}^{\prime }_{\boldsymbol{k}}\|_2^2+\frac {1}{k^6}\,\|{w}^{\prime \prime }_{\boldsymbol{k}}\|_2^2 }_{P} +\frac {2{\mathcal{R}} A}{\delta ^2}\left (I_B -I_T\right )\!, \end{equation}

in which

(3.17a) \begin{align} I_B= \int _0^\delta \left (\delta -Z \right )\left ( k^2\,|{w}_{\boldsymbol{k}}|^2-\frac {1}{k^4}\,{\rm Re}\{{w}^{\prime \prime }_{\boldsymbol{k}}{w}^*_{\boldsymbol{k}} \} \right )\mathrm{d}Z, \\[-12pt]\nonumber \end{align}

(3.17b) \begin{align} I_T=\int _{1-\delta }^{1} \left (Z-1+\delta \right )\left ( k^2|{w}_{\boldsymbol{k}}|^2-\frac {1}{k^4}\,{\rm Re}\{{w}^{\prime \prime }_{\boldsymbol{k}}{w}^*_{\boldsymbol{k}} \} \right )\mathrm{d}Z. \end{align}

3.3. Sufficient conditions for the spectral constraint

As a central step in proving Theorem1 using Proposition1, we establish in this subsection a closed-form sufficient condition on $\delta$ and $A$ , from $\phi$ given by (3.14), that guarantees that the spectral constraint (3.6) is satisfied. This amounts to ensuring the non-negativity of (3.16) for all $k\gt 0$ . To this end, we first present two lemmas, each providing a sufficient condition for the non-negativity of (3.16): one for the range $k\geq k_c$ , and another for $0\lt k\leq k_c$ , with $k_c\gt 0$ being an arbitrary threshold wavenumber. Then, by reconciling the ranges of large and small wavenumbers, we obtain a single condition to satisfy the spectral constraint.

3.3.1. For large wavenumbers $k\geq k_c$

Lemma 1 (Sufficient conditions for $Q_{\boldsymbol{k}}\geq 0$ , large $k$ ). Let $Q_{\boldsymbol{k}}$ be specified by ( 3.16 ) and ( 3.17 ), and let $k_c\gt 0$ be a lower threshold wavenumber. Suppose that $A\gt 0$ satisfies

(3.18) \begin{equation} {\mathcal{R}} A\leq \sqrt {\frac {2}{5}}\,k_c. \end{equation}

Then $Q_{\boldsymbol{k}}\geq 0$ for all $k\geq k_c$ .

Proof. We prove the lemma by constructing a lower bound on $Q_{\boldsymbol{k}}$ that is non-negative under the condition (3.18). The contribution of $\phi$ from (3.14) to $I_B$ and $I_T$ , as defined in (3.17a ) and (3.17b ), is symmetric. Hence apart from deriving a lower bound on $P$ , which has been defined in (3.16), we only prove upper and lower bounds on $I_B$ , since the identical estimates hold for $I_T$ . Using the boundary condition $w_{\boldsymbol{k}}(0)=0$ and integrating by parts, the $O(k^{-4})$ term in (3.17a ) can be written as

(3.19) \begin{equation} \begin{aligned} \int ^{\delta }_0 (\delta - Z)\,{{\rm Re}}\{ w''_{\boldsymbol{k}}w_{\boldsymbol{k}}^* \}\, \mathrm{d}Z &= \int ^{\delta }_0 \frac {1}{2} \left (\left |w_{\boldsymbol{k}}\right |^2 \right )' - (\delta -Z)\, |w'_{\boldsymbol{k}}|^2\, \mathrm{d}Z \\ &= \frac 12 |w_{\boldsymbol{k}}(\delta )|^2 - \int ^{\delta }_0 (\delta - Z )\,|w'_{\boldsymbol{k}}|^2\, \mathrm{d}Z. \end{aligned} \end{equation}

Then

(3.20) \begin{equation} \begin{aligned} I_B &= \int ^{\delta }_0 (\delta -Z) \left (k^2\, |w_{\boldsymbol{k}}|^2 + \frac {1}{k^4} |w'_{\boldsymbol{k}}|^2 \right ) \mathrm{d}Z -\frac {1}{2k^4}|w_{\boldsymbol{k}}(\delta )|^2 \\ &\leq \int ^{\delta }_0 (\delta -Z) \left (k^2\, |w_{\boldsymbol{k}}|^2 + \frac {1}{k^4} |w'_{\boldsymbol{k}}|^2 \right ) \mathrm{d}Z. \end{aligned} \end{equation}

Next, using that $|w|\leq \|w\|_\infty$ and $\left |w^{\prime }\right |\leq \|w^{\prime }\|_\infty$ , we get that

(3.21) \begin{equation} I_B\leq \int _{0}^{\delta } \left (\delta -Z\right )\left (k^2\,\|w_{\boldsymbol{k}}\|_\infty ^2+\frac {1}{k^4}\|w'_{\boldsymbol{k}}\|_\infty ^2\right )\mathrm{d}Z=\frac {\delta ^2}{2}\left (k^2\,\|w_{\boldsymbol{k}}\|_\infty ^2+\frac {1}{k^4}\|w'_{\boldsymbol{k}}\|_\infty ^2\right )\!. \end{equation}

On the other hand, the fundamental theorem of calculus implies

(3.22) \begin{equation} \left |w_{\boldsymbol{k}}(Z)\right |=\left |\int _0^Z w'_{\boldsymbol{k}}(\eta )\,\mathrm{d}\eta \right |\leq \int _0^Z \|w'_{\boldsymbol{k}}\|_{\infty }\,\mathrm{d}\eta =Z\,\|w'_{\boldsymbol{k}}\|_{\infty }, \end{equation}

hence from the identity in (3.20), $I_B$ is lower bounded as

(3.23) \begin{equation} I_B\geq -\frac {1}{2k^4}\left |w_{\boldsymbol{k}}(\delta )\right |^2\geq -\frac {\delta ^2}{2k^4}\|w'_{\boldsymbol{k}}\|_{\infty }^2. \end{equation}

Since $I_T$ is upper bounded by the right-hand side of (3.21), inserting (3.21) and (3.23) into (3.16) yields a sufficient condition for $Q_{\boldsymbol{k}}\geq 0$ :

(3.24) \begin{equation} P - {\mathcal{R}} A \left (k^2\,\|w_{\boldsymbol{k}}\|_\infty ^2+\frac {2}{k^4}\|w'_{\boldsymbol{k}}\|_\infty ^2\right ) \geq 0. \end{equation}

To proceed, we estimate $P$ from below. Denote $w_{\boldsymbol{k}}(Z)=w_{r}(Z)+\mathrm{i}\,w_{i}(Z)$ , with $w_r$ and $w_i$ being real functions. As $w_{\boldsymbol{k}}(1)=w_{\boldsymbol{k}}(0)=0$ , the mean value theorem guarantees the existence of points $Z_i \in [0,1 ]$ such that ${w}^{\prime }_{r}(Z_r) ={w}^{\prime }_{i}(Z_i)=0$ . Hence for any $Z\in [0,1]$ , the pointwise estimate holds that

(3.25) \begin{equation} \begin{aligned} \left |w'_{\boldsymbol{k}}(Z)\right |^2&=\left |w'_{r}(Z)\right |^2+\left |w'_{i}(Z)\right |^2\\ &=2\int _{Z_r}^Z {w_{r}^{\prime }}(\eta )w_{r}^{\prime \prime }(\eta )\,\mathrm{d}\eta +2\int _{Z_i}^Z {w_{i}^{\prime }}(\eta )w_{i}^{\prime \prime }(\eta )\,\mathrm{d}\eta \\ &\leq 2\int _0^1 (\left |w_r'(\eta )\right |\left |w_{r}^{\prime \prime }(\eta )\right |+\left |w_i'(\eta )\right |\left |w_{i}^{\prime \prime }(\eta )\right |)\, \mathrm{d}\eta . \end{aligned} \end{equation}

Taking the supremum over $Z$ and applying the Cauchy–Schwarz inequality then gives

(3.26) \begin{equation} \begin{aligned} \|w'_{\boldsymbol{k}}\|_{\infty }^{2}&=\sup _{Z}\left |w_{\boldsymbol{k}}^{\prime }(Z)\right |^2 \leq 2\left (\|w_r'\|_2\,\|w_r''\|_2+\|w_i'\|_2\,\|w_i''\|_2 \right )\\ &\leq 2\,\sqrt {\|w_r'\|_2^2+\|w_i'\|_2^2}\,\sqrt {\|w_r''\|_2^2+\|w_i''\|_2^2}\\ &=2\,\|w_{\boldsymbol{k}}^{\prime }\|_2\, \|w_{\boldsymbol{k}}^{\prime \prime }\|_2. \end{aligned} \end{equation}

Similarly, one obtains

(3.27) \begin{equation} \|w_{\boldsymbol{k}}\|_{\infty }^{2}\leq 2\,\|w_{\boldsymbol{k}}\|_2\, \|w_{\boldsymbol{k}}^{\prime }\|_2. \end{equation}

Both of the above estimates are instances of the one-dimensional Gagliardo–Nirenberg interpolation inequality (Nirenberg Reference Nirenberg1959).

By Young’s inequality, together with inequalities (3.26) and (3.27), the quantity $P$ in (3.16) is bounded from below as

(3.28) \begin{align} P&=k^6\,\|w_{\boldsymbol{k}}\|_2^2+\frac {2}{5}\|w'_{\boldsymbol{k}}\|_2^2+\frac {8}{5}\|w'_{\boldsymbol{k}}\|_2^2+\frac {1}{k^6}\|w''_{\boldsymbol{k}}\|_2^2 \\[-12pt]\nonumber \end{align}

(3.29) \begin{align} &\geq 2k^3\sqrt {\frac {2}{5}}\,\|w_{\boldsymbol{k}}\|_2\,\|w'_{\boldsymbol{k}}\|_2+\frac {4}{k^3}\sqrt {\frac {2}{5}}\,\|w'_{\boldsymbol{k}}\|_2\,\|w''_{\boldsymbol{k}}\|_2 \nonumber \\ &\geq \sqrt {\frac {2}{5}}\left (k^3\,\|w_{\boldsymbol{k}}\|_\infty ^2+\frac {2}{k^3}\|w'_{\boldsymbol{k}}\|_\infty ^2\right )\!. \end{align}

Next, substituting (3.29) into (3.24) gives the condition

(3.30) \begin{equation} Q_{\boldsymbol{k}}\geq \left (\sqrt {\frac {2}{5}}\,k-{\mathcal{R}} A\right )\left (k^2\,\|w_{\boldsymbol{k}}\|_\infty ^2+\frac {2}{k^4}\|w'_{\boldsymbol{k}}\|_\infty ^2\right ) \geq 0, \end{equation}

which is satisfied for all $k\geq k_c$ if and only if (3.18) holds.

3.3.2. For small wavenumbers $0\lt k\leq k_c$

Lemma 2 (Sufficient conditions for $Q_{\boldsymbol{k}}\geq 0$ , small $k$ ). Let $Q_{\boldsymbol{k}}$ be specified by ( 3.16 ) and ( 3.17 ), let $k_c\gt 0$ be an upper threshold wavenumber, and let $a\in (0,1)$ be an arbitrary weight. Suppose that the positive pair $(A,\delta )$ satisfies

(3.31) \begin{equation} {\mathcal{R}} A\leq \min \left \{\frac {6a}{k_c^2\delta },\ \sqrt {\frac {5}{k_c\delta }}\left (\frac {2-2a}{3}\right )^{{1}/{4}} \right \}\!. \end{equation}

Then $Q_{\boldsymbol{k}} \geq 0$ for all $0\lt k\leq k_c$ .

Proof. First, given that $w_{\boldsymbol{k}}(0)=0$ , we can use the fundamental theorem of calculus and Hölder’s inequality to write (3.22) and also that

(3.32) \begin{equation} \left | w_{\boldsymbol{k}}(Z)\right | =\left | \int ^{Z}_0 w_{\boldsymbol{k}}^{\prime }(\eta )\,\mathrm{d}\eta \right | \leq \sqrt {Z}\,\|w_{\boldsymbol{k}}^{\prime } \|_2. \end{equation}

Then, substituting (3.22) and (3.32) into (3.17a ), integrating in $Z$ and using the Cauchy–Schwarz inequality gives

(3.33) \begin{align} I_B &\leq k^2\, \| w'_{\boldsymbol{k}}\|_2^2 \int ^{\delta }_0 (\delta - Z)Z\, \mathrm{d}Z + \frac {1}{k^4} \| w'_{\boldsymbol{k}}\|_\infty \int ^{\delta }_0 (\delta -Z)Z\, |w''_{\boldsymbol{k}}|\,\mathrm{d}Z \nonumber \\ &\leq \frac 16 \delta ^3k^2\, \| w'_{\boldsymbol{k}}\|_2^2 + \frac {\delta ^{5/2}}{\sqrt {30}\, k^4} \|w'_{\boldsymbol{k}} \|_\infty\, \|w''_{\boldsymbol{k}}\|_2. \end{align}

By symmetry, $I_T$ is bounded from above by the right-hand side of (3.33). A similar procedure yields

(3.34) \begin{equation} I_B\geq -\frac {1}{k^4} \| w'_{\boldsymbol{k}}\|_\infty \int ^{\delta }_0 (\delta -Z)Z\, |w''_{\boldsymbol{k}}|\,\mathrm{d}Z \geq -\frac {\delta ^{5/2}}{\sqrt {30}\, k^4} \|w'_{\boldsymbol{k}} \|_\infty\, \|w''_{\boldsymbol{k}}\|_2. \end{equation}

Next, consider $P$ . Let $a,b\in (0,1)$ be weighting parameters. By Young’s inequality and (3.26), the non-negative term in (3.16) can be written as

(3.35) \begin{align} P&=k^6\,\|w_{\boldsymbol{k}}\|_2^2+2a\,\|w'_{\boldsymbol{k}}\|_2^2+2(1-a)\,\|w'_{\boldsymbol{k}}\|_2^2+\frac {1-b}{k^6}\|w''_{\boldsymbol{k}}\|_2^2+\frac {b}{k^6}\|w''_{\boldsymbol{k}}\|_2^2 \nonumber \\ &\geq 2a\,\|w'_{\boldsymbol{k}}\|_2^2+\frac {2}{k^3}\sqrt {2(1-a)(1-b)}\,\|w'_{\boldsymbol{k}}\|_2\,\|w''_{\boldsymbol{k}}\|_2+\frac {b}{k^6}\|w''_{\boldsymbol{k}}\|_2^2 \nonumber \\ &\geq 2a\,\|w'_{\boldsymbol{k}}\|_2^2+\frac {1}{k^3}\sqrt {2(1-a)(1-b)}\,\|w'_{\boldsymbol{k}}\|_{\infty }^2+\frac {b}{k^6}\|w''_{\boldsymbol{k}}\|_2^2 \nonumber \\ &\geq 2a\,\|w'_{\boldsymbol{k}}\|_2^2+\frac {2}{k^{9/2}}b^{{1}/{2}}(1-b)^{{1}/{4}}(2-2a)^{{1}/{4}}\,\|w'_{\boldsymbol{k}}\|_{\infty }\,\|w''_{\boldsymbol{k}}\|_2. \end{align}

Here, we fix $b= 2/3$ to maximise the pre-factor $b^{1/2}(1-b)^{1/4}$ . Substituting estimates (3.33)–(3.35) into (3.16) then yields

(3.36) \begin{align} Q_{\boldsymbol{k}}\geq \left (2a-\frac {1}{3} k^2\delta {\mathcal{R}} A\right )\|w'_{\boldsymbol{k}}\|_2^2+\frac {2}{k^{9/2}}\left [\left (\frac {2}{3}\right )^{{3}/{4}}(1-a)^{{1}/{4}}-2\sqrt {\frac {k\delta }{30}}\,{\mathcal{R}} A\right ]\|w'_{\boldsymbol{k}}\|_{\infty }\,\|w''_{\boldsymbol{k}}\|_2. \end{align}

It suffices to establish $Q_{\boldsymbol{k}}\geq 0$ for all $k\in (0,k_c]$ , provided that both terms on the right-hand side of (3.36) are non-negative on this interval. This requirement is equivalent to the condition (3.31).

3.3.3. Reconciling the two ranges

Aided by Lemmas1 and 2, we are now in a position to state the condition for the spectral constraint to be satisfied at all wavenumbers.

Lemma 3 (Sufficient conditions for spectral constraint). Let $\phi$ be defined by ( 3.14 ). Suppose that the positive pair $(A,\delta )$ in ( 3.14 ) satisfies

(3.37) \begin{align} {\mathcal{R}} A\leq c_{1}\,\delta ^{-{1}/{3}}, \quad \mathrm{where}\quad c_{{1}}=\frac {\sqrt {5}}{3}\left (\frac {3}{10}\sqrt {557}-\frac {3}{2}\sqrt {5}\right )^{{1}/{3}}=1.156. \end{align}

Then $(A,\delta )$ satisfies the spectral constraint.

Proof. If we equate the right-hand sides of (3.18) and (3.31), then solving the resulting two equations for $a$ and $k_c$ gives

(3.38) \begin{equation} a=\frac {5\sqrt {2785}-125}{216},\quad k_c=\frac {5}{3\sqrt {2}}\left (\frac {3}{10}\sqrt {557}-\frac {3}{2}\sqrt {5}\right )^{{1}/{3}}\delta ^{-{1}/{3}}. \end{equation}

Given (3.38), both conditions (3.18) and (3.31), for large and small $k$ , respectively, become equivalent to (3.37). Therefore, when (3.37) holds, $Q_{\boldsymbol{k}}\geq 0$ for all $k\gt 0$ .

In Grooms & Whitehead (Reference Grooms and Whitehead2014), the authors prove the non-negativity of a spectral constraint identical to (3.7), with an alternative strategy to the one in § 3.3. In contrast to Grooms & Whitehead (Reference Grooms and Whitehead2014), we do not examine the Green’s function solution of the diagnostic equation (2.14). Instead, we have demonstrated that integral estimates directly on the quantities in $Q_{\boldsymbol{k}}$ suffice to prove non-negativity. Even if our choice of $\phi (Z)$ was piecewise linear, as in Grooms & Whitehead (Reference Grooms and Whitehead2014), our strategy would work in the same way.

3.3.4. Optimality of the estimate

Before proving Theorem1, we demonstrate the optimality of the exponent $-1/3$ in (3.37) by establishing a necessary condition for the spectral constraint, using the single-mode velocity ansatz

(3.39a) \begin{equation} {w}_{\boldsymbol{k}}^{\textit{ans}}=\begin{cases} w_{k_a}^{\textit{ans}}(Z), &\boldsymbol{k}=(\pm k_a,0),\\[5pt] 0,& \boldsymbol{k}\neq (\pm k_a,0), \end{cases} \end{equation}

in which $k_a\gt 0$ is an arbitrary wavenumber, and

(3.39b) \begin{equation} w_{k_a}^{\textit{ans}}=\begin{cases} 0,& 0\leq Z\lt 1-\delta ,\\[11pt] \dfrac {1}{\delta ^5}\left (Z-1+\delta \right )^3\left (Z-1\right )\left (Z-1-\dfrac {\delta }{3}\right )\!, & 1-\delta \leq Z\leq 1. \end{cases} \end{equation}

It is straightforward to check that $w_{k_a}^{{ans}}$ satisfies the required boundary conditions and is continuously differentiable. Inserting this ansatz into (3.16) and rearranging gives

(3.40) \begin{equation} Q_{k_a}^{\textit{ans}} =\frac {64}{105\delta }\left (\frac {1}{k_a^6\delta ^2}+\frac {1}{12}+\frac {13}{4752}k_a^6\delta ^2\right )-\frac {\mathcal{R}A}{27k_a\delta }\left (\frac {1}{k_a^3\delta }+\frac {19}{330}k_a^3\delta \right ). \end{equation}

The condition $Q_{k_a}^{\textit{ans}}\geq 0$ is necessary for the spectral constraint to be satisfied. Noting that

(3.41) \begin{equation} \frac {1}{k_a^6\delta ^2}+\frac {1}{12}+\frac {13}{4752}k_a^6\delta ^2\lt \frac {1}{k_a^6\delta ^2}+\frac {7907}{75240}+\frac {13}{4752}k_a^6\delta ^2=\left (\frac {1}{k_a^3\delta }+\frac {19}{330}k_a^3\delta \right )\left (\frac {1}{k_a^3\delta }+\frac {65}{1368}k_a^3\delta \right )\!, \end{equation}

substituting this estimate into (3.40) and rearranging yields

(3.42) \begin{equation} Q_{k_a}^{\textit{ans}} \lt \frac {64}{105\delta }\left (\frac {1}{k_a^3\delta }+\frac {19}{330}k_a^3\delta \right )\left (\frac {1}{k_a^3\delta }+\frac {65}{1368}k_a^3\delta -\frac {35\mathcal{R}A}{576k_a}\right )\!. \end{equation}

A necessary condition for $Q_{k_a}^{\textit{ans}}\geq 0$ is the positivity of the sign-indefinite factor on the right-hand side, namely

(3.43) \begin{equation} \mathcal{R}A\lt \frac {576}{35}\left (\frac {1}{k_a^2\delta }+\frac {65}{1368}k_a^4\delta \right )\!. \end{equation}

We now specify $k_a= (684/65 )^{1/6}\delta ^{-1/3}$ to minimise the expression on the right-hand side, which yields the following necessary condition for the spectral constraint:

(3.44) \begin{equation} \mathcal{R}A\lt \frac {144}{7}\left ( \frac {78}{475}\right )^{{1}/{3}}\delta ^{-{1}/{3}}=11.27\,\delta ^{-{1}/{3}}. \end{equation}

Therefore, given the form of $\phi$ in (3.14), and assuming a priori that $\delta$ decreases as $\mathcal{R}$ increases beyond a certain threshold, no sufficient condition for the spectral constraint can exist in the form ${\mathcal{R}} A\lesssim \delta ^{-d}$ with the exponent $d\gt 1/3$ .

3.4. Proof of Theorem1

The final step towards Theorem1 is to specify $\mathcal{R}$ -dependent $A$ and $\delta$ that satisfy the condition of Lemma3. For algebraic simplicity, let

(3.45) \begin{equation} {\mathcal{R}}_{1}:=\frac {5\times2^{{7}/{3}}c_{1}}{3}=9.706 \end{equation}

be an a posteriori threshold parameter, where $c_{1}$ is given in (3.37). We then choose

(3.46) \begin{equation} \delta =\frac {1}{2}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{1}}\right )^{-{3}/{4}},\quad A=\frac {3}{20}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{1}}\right )^{-{3}/{4}}. \end{equation}

The requirement $\delta \leq 1/2$ imposes a lower threshold on $\mathcal{R}$ above which these choices are valid, namely ${\mathcal{R}}\geq {\mathcal{R}}_{1}$ . Substituting (3.46) into (3.15) yields the explicit expression

(3.47) \begin{equation} B_{1}= \frac {3}{25}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{1}}\right )^{-{3}/{4}}-\frac {1}{20}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{1}}\right )^{-{3}/{2}},\quad \forall {\mathcal{R}}\geq {\mathcal{R}}_{1}. \end{equation}

Since the choices of $\delta$ and $A$ satisfy the condition of Lemma3, from Proposition1, $B_1$ provides a lower bound on $\langle \varTheta \rangle _{V,t}$ . For small $\mathcal{R}$ , the unconstrained global maximum of (3.15), $1/12$ , is attained at $\delta =1/2$ and $A=1/4$ . This combination is feasible for ${\mathcal{R}}\leq 3{\mathcal{R}}_{1}/5$ , as determined by (3.37). Since $1/12$ is also the uniform upper bound, it follows that $\langle \varTheta \rangle _{V,t}=1/12$ in this regime, and the feasibility condition provides a sufficient condition for nonlinear stability of the conductive state. For ${\mathcal{R}}\gt 3{\mathcal{R}}_{1}/5$ , the spectral constraint (3.37) becomes active, and we choose

(3.48) \begin{equation} \delta =\sqrt {A}=\frac {1}{2}\left (\frac {5{\mathcal{R}}}{3 {\mathcal{R}}_{1}}\right )^{-{3}/{7}}, \end{equation}

which leads to the bound

(3.49) \begin{equation} B_{1}= \frac {1}{4}\left (\frac {5{\mathcal{R}}}{3{\mathcal{R}}_{1}}\right )^{-{6}/{7}}-\frac {1}{6}\left (\frac {5{\mathcal{R}}}{3{\mathcal{R}}_{1}}\right )^{-{9}/{7}},\quad \forall \mathcal{R}\geq \frac {3}{5}\mathcal{R}_{1}. \end{equation}

It remains effective up to ${\mathcal{R}}=1.976\,{\mathcal{R}}_{1}$ , beyond which (3.47) provides a sharper analytical bound. Theorem1 is therefore proven by substituting the value of $\mathcal{R}_1$ .

As demonstrated in Appendix C, the conductive state (2.12) is linearly stable for $\mathcal{R}\leq \mathcal{R}_c=71.75$ , hence the range of validity for (3.47) covers values of $\mathcal{R}$ above the linear onset of convection. However, although a lower bound is provided by $({3/5})\mathcal{R}_1$ , the nonlinear critical $\mathcal{R}$ remains unknown, and may be significantly lower than the linear threshold. For instance, in non-rotating IHC with stress-free and isothermal boundary conditions, the nonlinear critical Rayleigh number can be 40 % smaller than the linear critical value (Goluskin Reference Goluskin2016). Motivated by this, we have also proved bounds valid for ${\mathcal{R}} \lt \mathcal{R}_c$ . Lower bounds for $\langle \varTheta \rangle _{V,t}$ across different ranges of $\mathcal{R}$ are presented in figure 3. Ideally, one seeks the maximal lower bound, yet the maximisation of (3.15) under (3.37) can only be solved numerically. By using (3.47) and (3.49), we obtain analytical bounds that are at most 2.4 % below the numerical maximum, denoted by $B^{\textit{num}}_{1}$ .

For $\mathcal{R}\geq \mathcal{R}_{1}$ , $A$ and $\delta$ are chosen as in (3.46) so that the bound asymptotically approaches the maximal lower bound $B^{\textit{num}}_{1}$ , although the QG regimes extend only to finite $\mathcal{R}$ . This is shown as follows. By construction, for $\mathcal{R}\geq \mathcal{R}_{1}$ we have

(3.50)

with the combination (3.46) being the maximiser. Hence, compared with (3.47),

(3.51) \begin{equation} B_{1}\lt B^{\textit{num}}_{1}\lt \big [1+O(\mathcal{R}^{-3/4})\big ]B_{1}, \end{equation}

which implies that (3.47) is equivalent to the maximal lower bound on leading order. For $\mathcal{R}\leq \mathcal{R}_{1}$ , $A$ and $\delta$ are chosen as in (3.48) so that the bound is tangent to $B^{\textit{num}}_1$ at $3\mathcal{R}_{1}/5$ .

In (3.47) and (3.49), the pre-factors of terms are determined by $\phi$ , while $\mathcal{R}_{1}$ arises from the particular estimates used to bound $Q_{\boldsymbol{k}}$ from below. As such, ${\mathcal{R}}_1$ is a product of the methodology and can be improved if sharper estimates on $Q_{\boldsymbol{k}}$ are possible.

Figure 3.

Lower bounds on $\langle \varTheta \rangle _{V,t}$ from (3.47) (blue solid line) and (3.49) (red dashed line), compared with the uniform upper bound $1/12$ (grey solid line) and the numerical maximum (black dots) of (3.15) subject to (3.37) and the condition $\delta \leq 1/2$ . Triangle markers denote the $\mathcal{R}$ value above which (3.47) and (3.49) are valid, with the red triangle corresponding to a lower threshold of nonlinear convective instability, and the star indicating the linear onset of convection.

4.1. Bounding framework

The same method as that outlined in § 3.1 is used to bound the mean convective heat flux $\langle {w\theta }\rangle _{V,t}$ from above. Since the time average of a quantity does not exceed the pointwise supremum, we have that

(4.1) \begin{equation} \langle {w\theta }\rangle _{V,t}=\limsup _{\tau \rightarrow \infty }\frac {1}{\tau }\int ^{\tau }_0 \left(\langle {w\theta }\rangle _V+\frac {\mathrm{d}\mathcal{V}}{\mathrm{d}t}\right)\mathrm{d}t\leq \sup \left ( \langle {w\theta }\rangle _V+\frac {\mathrm{d}\mathcal{V}}{\mathrm{d}t} \right )\leq B_{2}, \end{equation}

where $\mathcal{V}$ is defined by (3.3). It thereby suffices for $B_{2}$ to be an upper bound on $\langle {w\theta }\rangle _{V,t}$ provided that

(4.2) \begin{equation} Q:= B_{2}- \langle {w\theta }\rangle _{V}-\frac {\mathrm{d}\mathcal{V}}{\mathrm{d}t}\geq 0. \end{equation}

On this basis, we state the problem.

Proposition 2 (Bound on $\langle {w\theta }\rangle _{V,t}$ ). Let $\phi :[0,1] \rightarrow \mathbb{R}$ satisfy the boundary conditions ( 3.4 ), and let $\beta \gt 0$ . Then the upper bound $B_{2}$ on $\langle w \theta \rangle _{V,t}$ is given by

(4.3) \begin{equation} B_{2}:=\frac {\beta }{48}+\int _0^1 \left(\frac {1}{4\beta }{\phi '}^2-\frac {1}{2}\phi \right)\mathrm{d}Z, \end{equation}

provided that $\phi$ and $\beta$ further satisfy the spectral constraint

(4.4) \begin{equation} Q_{\boldsymbol{k}}\left \{w,\phi ,\beta \right \}\geq 0,\quad \forall k\gt 0,\quad \forall \left .w\,\right | \,{w}(0)={w}(1)=0, \end{equation}

in which

(4.5) \begin{equation} Q_{\boldsymbol{k}}:= k^6\,\|{w}_{\boldsymbol{k}}\|^2_2+2\,\|{w}^{\prime }_{\boldsymbol{k}}\|^2_2+\frac {1}{k^6}\|{w}^{\prime \prime }_{\boldsymbol{k}}\|^2_2+\frac {{\mathcal{R}}}{\beta }\int _0^1\left (\phi '-1\right )\left (k^2\left |w_{\boldsymbol{k}}\right |^2-\frac {1}{k^4}\,{\rm Re} \{{w}^{\prime \prime }_{\boldsymbol{k}}{w}^*_{\boldsymbol{k}} \}\right ) \mathrm{d}Z. \end{equation}

Proof. Similar to the demonstration of Proposition1, inserting (3.8) into (4.2), and expanding $w$ and $\theta$ in horizontal Fourier modes leads to the decomposition of $Q$ in (3.9), and the condition $Q\geq 0$ becomes two independent conditions:

(4.6a) \begin{align} Q_0=B_{2}+\int _0^1 (\beta \,|\varTheta ' |^2-\left (\phi '-\beta Z \right )\varTheta '+\phi)\,\mathrm{d}Z \geq 0, \\[-12pt]\nonumber \end{align}

(4.6b) \begin{align} Q_{\boldsymbol{k}}=\beta k^2\,\|{\theta }_{\boldsymbol{k}}\|^2_2+\int _0^1 (\phi '-1 )\,{Re} \{{w}_{\boldsymbol{k}}{\theta }_{\boldsymbol{k}}^* \}\, \mathrm{d}Z \geq 0. \end{align}

Since $\beta$ is positive, the integrand in $Q_0$ is convex in $\varTheta '$ . Then the worst-case value of $B_{2}$ is given by

(4.7) \begin{equation} B_{2} = \sup _{ \substack {\varTheta (Z)\\\varTheta (0)= \varTheta (1)=0 } }{\int _0^1 (-\beta \,|\varTheta '|^2+ (\phi '-\beta Z )\varTheta '-\phi )\,\mathrm{d}Z}, \end{equation}

the maximiser of which is solved as

(4.8) \begin{equation} \varTheta '_{\textit{opt}}=\frac {\phi '}{2\beta } - \frac {Z}{2} + \frac 14. \end{equation}

Substituting $\varTheta '_{\textit{opt}}$ back into $B_2$ yields the desired result.

Using the diagnostic relation (2.14) and integrating by parts, $Q_{\boldsymbol{k}}$ can be equivalently written as

(4.9) \begin{equation} \begin{aligned} Q_{\boldsymbol{k}}&=\frac {\beta }{{\mathcal{R}}^2}\left (k^6\,\|{w}_{\boldsymbol{k}}\|^2_2+2\,\|{w_{\boldsymbol{k}}^{\prime }}\|^2_2 +\frac {1}{k^6}\|{w_{\boldsymbol{k}}^{\prime \prime }}\|^2_2\right )\\ &\quad{}+\frac {1}{{\mathcal{R}} }\int _0^1\left (\phi '-1\right )\left (k^2\left |w_{\boldsymbol{k}}\right |^2-\frac {1}{k^4}\,{\rm Re}\{{w_{\boldsymbol{k}}^{\prime \prime }}{w_{\boldsymbol{k}}}^*\}\right ) \mathrm{d}Z, \end{aligned} \end{equation}

the non-negativity of which for all admissible $\boldsymbol{k}$ and $w_{\boldsymbol{k}}$ is satisfied by the spectral constraint. Hence the condition (4.2) is satisfied by $B_{2}$ as given in (4.3), together with the spectral constraint.

4.2. Ansatz

For the proof of Theorem2, a family of background fields $\phi$ different from those in § 3.1 is used. The background field is characterised by a quadratic top boundary layer, and is linear in the bulk with $\phi '=1$ :

(4.10) \begin{equation} \phi = \begin{cases} Z,&0\leq Z\lt 1-\delta ,\\[5pt] ({1}/{\delta ^2})\left (1-Z\right )\left (2\delta -\delta ^2+Z-1\right ), &1-\delta \leq Z\leq 1. \end{cases} \end{equation}

As illustrated in figure 4, $\phi$ is fully specified by the boundary layer thickness $\delta \in (0,1 ]$ .

Figure 4.

A sketch of the background field (4.10) with a quadratic boundary layer and linear bulk.

When the ansatz (4.10) is substituted into (4.3) and (4.5), we get

(4.11) \begin{equation} B_{2}=\frac {\beta }{48}+\frac {1}{3\beta \delta }-\frac {1}{4\beta }+\frac {\delta }{6}-\frac {1}{4} \end{equation}

and

(4.12) \begin{equation} Q_{\boldsymbol{k}}= \underbrace { k^6\,\|{w}_{\boldsymbol{k}}\|_2^2+2\,\|{w}^{\prime }_{\boldsymbol{k}}\|_2^2+\frac {1}{k^6}\|{w}^{\prime \prime }_{\boldsymbol{k}}\|_2^2 }_{P}-\frac {2{\mathcal{R}}}{\beta \delta ^2}I_T, \end{equation}

respectively. Therein, $I_T$ is the same as in (3.17b ).

4.3. Sufficient conditions for the spectral constraint

Similar to § 3.3, the proof of Theorem2 relies on a lemma presented in this subsection, which provides a closed-form sufficient condition on the parameter $\phi$ in (4.10) and on the balance parameter $\beta$ to ensure that the spectral constraint (4.4) is satisfied. Before stating this result, we again present two auxiliary lemmas giving sufficient conditions for the non-negativity of $Q_{\boldsymbol{k}}$ over different ranges of wavenumbers.

Lemma 4 (Sufficient conditions for $Q_{\boldsymbol{k}}\geq 0$ , large $k$ ). Let $Q_{\boldsymbol{k}}$ be specified by ( 3.17 ) and ( 4.12 ), and let $k_c\gt 0$ be a lower threshold wavenumber. Suppose that the positive parameter $\beta$ satisfies

(4.13) \begin{equation} \frac {{\mathcal{R}} }{\beta }\leq k_c. \end{equation}

Then $Q_{\boldsymbol{k}}\geq 0$ for all $k\geq k_c$ .

Proof. By applying Young’s inequality along with inequalities (3.26) and (3.27), $P$ in (4.12) is estimated from below as

(4.14) \begin{equation} \begin{aligned} P&=k^6\, \|w_{\boldsymbol{k}}\|_2^2+\|w_{\boldsymbol{k}}^{\prime }\|_2^2+\|w_{\boldsymbol{k}}^{\prime }\|_2^2+\frac {1}{k^6}\|w_{\boldsymbol{k}}^{\prime \prime }\|_2^2 \\ &\geq 2k^3\,\|w_{\boldsymbol{k}}\|_2\,\|w_{\boldsymbol{k}}^{\prime }\|_2+\frac {2}{k^3}\|w_{\boldsymbol{k}}^{\prime }\|_2\,\|w_{\boldsymbol{k}}^{\prime \prime }\|_2\\ &\geq k^3\, \|w_{\boldsymbol{k}}\|^2_{\infty }+\frac {1}{k^3}\|w_{\boldsymbol{k}}^{\prime }\|^2_{\infty }. \end{aligned} \end{equation}

Substituting (4.14) and the upper bound for $I_T$ (3.21) into (4.12) yields the lower estimate on $Q_{\boldsymbol{k}}$ :

(4.15) \begin{equation} Q_{\boldsymbol{k}}\geq \left (k-\frac {{\mathcal{R}}}{\beta }\right )\left (k^2\, \|w_{\boldsymbol{k}}\|^2_{\infty }+\frac {1}{k^4}\|w_{\boldsymbol{k}}^{\prime }\|_{\infty }^2\right )\!. \end{equation}

Under condition (4.13), the right-hand side of (4.15) is non-negative for all $k\geq k_c$ .

On the other hand, for smaller wavenumbers, the following lemma applies.

Lemma 5 (Sufficient conditions for $Q_{\boldsymbol{k}}\geq 0$ , small $k$ ). Let $Q_{\boldsymbol{k}}$ be specified by ( 3.17 ) and ( 4.12 ), let $k_c\gt 0$ be an upper threshold wavenumber, and let $a\in (0,1)$ be an arbitrary number. Suppose that the positive pair $(\beta ,\delta )$ satisfies

(4.16) \begin{equation} \frac {{\mathcal{R}}}{\beta }\leq \min \left \{ \frac {6a}{k_c^2 \delta },\ 2\,\sqrt {\frac {5}{k_c \delta }}\left (\frac {2-2a}{3}\right )^{{1}/{4}} \right \}\!. \end{equation}

Then $Q_{\boldsymbol{k}}\geq 0$ for all $0\lt k\leq k_c$ .

Proof. Applying the estimates (3.33) and (3.35) to (4.12) yields the lower bound

(4.17) \begin{equation} \begin{aligned} Q_{\boldsymbol{k}} &\geq \left (2a-\frac {k^2\delta {\mathcal{R}} }{3\beta }\right )\|w_{\boldsymbol{k}}^{\prime }\|_2^2+\frac {2}{k^{9/2}}\left [\left (\frac {2}{3}\right )^{{3}/{4}}(1-a)^{{1}/{4}}-\sqrt {\frac {k\delta }{30}}\frac {{\mathcal{R}} }{\beta }\right ]\|w_{\boldsymbol{k}}^{\prime }\|_{\infty }\,\|w_{\boldsymbol{k}}^{\prime \prime }\|_2. \end{aligned} \end{equation}

This lower estimate is non-negative provided that the pre-factors of both terms on the right-hand side are non-negative for all $k\in (0, k_c ]$ . This condition is equivalent to (4.16), which suffices for the non-negativity of $Q_{\boldsymbol{k}}$ .

Finally, taking the two auxiliary lemmas together, we arrive at a sufficient condition for the spectral constraint.

Lemma 6 (Sufficient conditions for spectral constraint). Let $\phi$ be defined by ( 4.10 ). Suppose that the positive pair $(\beta ,\delta )$ satisfies

(4.18) \begin{equation} \frac {{\mathcal{R}} }{\beta }\leq c_{2}\, \delta ^{-{1}/{3}},\quad \mathrm{where}\quad c_{2}=\left [\frac {20}{9}\left (\sqrt {154}-10\right )\right ]^{{1}/{3}}=1.750. \end{equation}

Then $(\beta ,\delta )$ satisfies the spectral constraint.

Proof. If we equate the right-hand sides of (4.13) and (4.16), then solving the resulting two equations for $a$ and $k_c$ gives

(4.19) \begin{equation} a=\frac {10}{27}\left (\sqrt {154}-10\right )\!,\quad k_c=\left [\frac {20}{9}\left (\sqrt {154}-10\right )\right ]^{{1}/{3}}\delta ^{-{1}/{3}}. \end{equation}

Given (4.19), both conditions of Lemmas4 and 5, for large and small $k$ respectively, become equivalent to (4.18). Therefore, when (4.18) holds, $Q_{\boldsymbol{k}} \geq 0$ for all $k\gt 0$ .

The velocity ansatz (3.39) may also be employed to demonstrate the optimality of the exponent $-1/3$ in the condition (4.18). Substituting (3.39) into (4.12), applying the estimate (3.41) and rearranging, we obtain

(4.20) \begin{equation} \begin{aligned} Q_{k_a}^{\textit{ans}} &= \frac {64}{105\delta }\left (\frac {1}{k_a^6\delta ^2}+\frac {1}{12}+\frac {13}{4752}k_a^6\delta ^2\right )-\frac {\mathcal{R}}{27\beta k_a\delta }\left (\frac {1}{k_a^3\delta }+\frac {19}{330}k_a^3\delta \right )\\ &\lt \frac {64}{105\delta }\left (\frac {1}{k_a^3\delta }+\frac {19}{330}k_a^3\delta \right )\left (\frac {1}{k_a^3\delta }+\frac {65}{1368}k_a^3\delta -\frac {35\mathcal{R}}{576\beta k_a}\right )\!. \end{aligned} \end{equation}

With $k_a= (684/65 )^{1/6}\delta ^{-1/3}$ specified, a necessary condition for the spectral constraint to be satisfied is then

(4.21) \begin{equation} \frac {{\mathcal{R}} }{\beta }\lt \frac {576}{35}\left (\frac {1}{k_a^2\delta }+\frac {65}{1368}k_a^4\delta \right )=\frac {144}{7}\left ( \frac {78}{475}\right )^{{1}/{3}}\delta ^{-{1}/{3}}. \end{equation}

Therefore, no sufficient condition for the spectral constraint can take the form ${\mathcal{R}}/{\beta }\lesssim \delta ^{-d}$ with $d\gt 1/3$ , if $\delta$ decreases as $\mathcal{R}$ increases beyond a certain threshold.

4.4. Proof of Theorem2

To complete the demonstration of Theorem2, we define a posteriori

(4.22) \begin{equation} {\mathcal{R}}_{2} := 2^{3}c_{2}=14.00, \end{equation}

where $c_{2}$ is defined in (4.18), and choose in Lemma6

(4.23) \begin{equation} \beta =8\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{{3}/{5}},\quad \delta =\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{-{6}/{5}}, \end{equation}

which satisfy the condition (4.18) and, based on Proposition2, yield the upper bound on $\langle {w\theta } \rangle _{V,t}$ :

(4.24) \begin{equation} B_{2}=\frac {5}{24}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{{3}/{5}}-\frac {1}{4}-\frac {1}{32}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{-{3}/{5}}+\frac {1}{6}\left (\frac {{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{-{6}/{5}} ,\quad \forall {\mathcal{R}}\geq {{\mathcal{R}}_{2}}. \end{equation}

This bound is valid for ${\mathcal{R}}\geq {\mathcal{R}}_{2}$ , where the requirement $\delta \leq 1$ is satisfied. For small $\mathcal{R}$ , an unconstrained global minimum of (4.11), $0$ , is achieved at $\delta = 1$ and $\beta = 2$ . By the condition (4.18), this choice is feasible up to ${\mathcal{R}} = {\mathcal{R}}_{2}/4$ . Since 0 is also the uniform lower bound, it follows that $\langle w\theta \rangle _{V,t}=0$ in this regime. For ${\mathcal{R}} \gt {\mathcal{R}}_2/4$ , we then choose

(4.25) \begin{equation} \delta =\sqrt {\frac {2}{\beta }}=\left (\frac {4{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{-{3}/{7}}, \end{equation}

which gives the bound

(4.26) \begin{equation} B_{2}=\frac {1}{24}\left (\frac {4{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{{6}/{7}}-\frac {1}{4}+\frac {1}{3}\left (\frac {4{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{-{3}/{7}}-\frac {1}{8}\left (\frac {4{\mathcal{R}}}{{\mathcal{R}}_{2}}\right )^{-{6}/{7}} ,\quad \forall \mathcal{R}\geq \frac {1}{4}\mathcal{R}_{2}. \end{equation}

This bound becomes suboptimal to (4.24) for ${\mathcal{R}} \geq 2.160\,{\mathcal{R}}_{2}$ . With $\mathcal{R}_{2}$ specified by (4.22), Theorem2 is then proved.

Equation (4.24) is suboptimal once it exceeds the uniform upper bound $\langle {w\theta }\rangle _{V,t}\leq 1/2$ , which occurs at ${\mathcal{R}}=8.376\,{\mathcal{R}}_{2}=117.2$ . The upper bounds on $\langle w\theta \rangle _{V,t}$ for different ranges of $\mathcal{R}$ are displayed in figure 5, where the analytical bounds lie at most 10 % above the numerical solution of minimising (4.11) subject to the constraint (4.18) and $\delta \leq 1$ .

Figure 5.

Upper bounds on $\langle w\theta \rangle _{V,t}$ from (4.24) (blue solid line) and (4.26) (red dashed line), compared with the uniform bounds $0$ and $1/2$ (grey solid lines), along with the numerical minimum (black dots) of (4.11) subject to the condition (4.18) and $\delta \leq 1$ . The triangle markers denote the $\mathcal{R}$ value above which (4.24) and (4.26) are valid, and the star indicates the linear onset of convection.

Figure 5. Long description

A line graph with the horizontal axis labeled R and the vertical axis labeled B 2. The graph features two data lines: a blue solid line and a red dashed line. The blue solid line represents the upper bounds from equation 4.24, while the red dashed line represents the upper bounds from equation 4.26. Black dots indicate the numerical minimum of equation 4.11 subject to the condition 4.18. Triangle markers denote the value above which equations 4.24 and 4.26 are valid, and a star indicates the linear onset of convection. The graph includes annotations for specific values such as R c equals 71.75, R 2 equals 14.00, and 117.2. The blue solid line shows a trend of increasing B 2 with increasing R, following a power law of approximately R to the power of 3/5. The red dashed line also shows an increasing trend but at a different rate. Grey solid lines represent uniform bounds.