2. Set-up, control and response parameters

Our CC set-up is a vertically aligned cylindrical annulus with top and bottom solid walls, a cooled inner sidewall and a heated outer sidewall, all rotating clockwise about the vertical axis at constant angular velocity $\varOmega \geq 0$ (see figure 1). The container has height $H$ , outer radius $R$ and gap $L$ between the inner and outer vertical walls. The outer wall is at temperature $\theta _+$ and the inner wall at $\theta _-$ , so that $\varDelta \equiv \theta _+-\theta _-\gt 0$ . The horizontal walls are perfectly insulated (adiabatic), and the velocity satisfies no-slip boundary conditions ${\boldsymbol{u}}=0$ on all walls. The two aspect ratios are $\varGamma _{\phi }\equiv 2\pi R/L=4\pi$ and $\varGamma _H\equiv H/L=1$ . The fluid has fixed thermal expansion coefficient $\alpha$ , kinematic viscosity $\nu$ and thermal diffusivity $\kappa$ . The total acceleration is $\hat g \equiv ({{g}_z^{2}+{g}_\varOmega ^{2}})^{1/2}$ , where ${g}_\varOmega \equiv \varOmega ^2R$ and $g_z$ is gravitational acceleration. The non-dimensional control parameters are the Rayleigh number ${\textit{Ra}}$ , Froude number ${\textit{Fr}}$ and Prandtl number $Pr$ :

(2.1) \begin{equation} {{\textit{Ra}}}\equiv \alpha \hat g L^3 \varDelta /(\kappa \nu ),\quad {{\textit{Fr}}}\equiv {g_\varOmega }/{g_z}, \quad Pr\equiv {\nu }/{\kappa }=4.3 \text{ (water at about } 40\,^{\circ }\text{C)}. \end{equation}

For fixed $\varGamma _\phi$ and $\varGamma _H$ , the CC system has four independent control parameters: ${\textit{Ra}}$ , $Pr$ , ${\textit{Fr}}$ and either the Rossby number $Ro$ or the Ekman number ${Ek}$ . However, for given fluid (so that all fluid properties and $Pr$ are fixed) and container geometry (including the aspect ratios $\varGamma _\phi$ and $\varGamma _H$ , and the physical outer radius of the annulus $R$ , so that the quantity $A\equiv {8\pi ^2\varGamma _\phi ^{-2}}({g_zR^3/(\nu \kappa )})^{1/2}$ is also fixed), the CC system has only two independent parameters: one for the thermal driving, ${\textit{Ra}}$ , and one for the rotation strength, here chosen as ${\textit{Fr}}$ . The other rotation parameters become dependent parameters, as ${Ro} \equiv {\sqrt {\alpha \hat g L \varDelta }}/({2 \varOmega L})={A^{-1}}\sqrt {{{{\textit{Ra}}}}/{{{\textit{Fr}}}}}$ and ${{Ek}} \equiv {\nu }/({2 \varOmega L^2})={A^{-1}}\sqrt {{\textit{Pr}}/{{{\textit{Fr}}}}}$ .

The main response parameters in the system are the Nusselt and Reynolds numbers:

(2.2) \begin{equation} {\textit{Nu}} = {r\kappa ^{-1}\varDelta ^{-1} \log (1-2\pi /\varGamma _\phi ) \bigl[\langle u_r \theta \rangle _{\varphi ,z,t}-\kappa \partial _r \langle \theta \rangle _{\varphi ,z,t}\bigr]}, \quad \textit{Re} = L\,\nu ^{-1}\langle {\boldsymbol{u}}\boldsymbol{\cdot }{\boldsymbol{u}}\rangle ^{1/2}, \end{equation}

where $\boldsymbol{u}$ is the velocity field, $\theta$ the temperature, reduced by the arithmetic mean of the smallest ( $\theta _-$ ) and largest ( $\theta _+$ ) boundary temperatures, and $\langle \boldsymbol{\cdot }\rangle$ the time and volume average.

Figure 2.

Phase diagram of the conducted DNS in the $({{\textit{Ra}}}, k_i)$ parameter space for ( $a$ ) ${{\textit{Fr}}} = 10$ and ( $b$ ) ${{\textit{Fr}}} = 100$ . Different symbol shapes indicate different ${\textit{Ra}}$ , and colours denote the final flow state after 3000 time units: pink – flow with a persistent coherent roll structure, where initial ( $k_i$ ) and final ( $k$ ) roll numbers coincide, $k = k_i$ , and $\sigma _{{\textit{Nu}}}\leq 0.1$ , $\sigma _{\textit{Re}}\leq 0.1$ , with $\sigma _{{\textit{Nu}}}^2\equiv \langle ({\textit{Nu}}_t-{\textit{Nu}})^2\rangle _t$ and $\sigma _{\textit{Re}}^2\equiv \langle ({\textit{Re}}_t-Re)^2\rangle _t$ , where ${\textit{Nu}}_t$ and ${\textit{Re}}_t$ are the instantaneous Nusselt and Reynolds numbers and $\langle \boldsymbol{\cdot }\rangle _t$ represents time averaging; orange – flow with $k = k_i$ , but $\sigma _{{\textit{Nu}}}\gt 0.1$ or $\sigma _{\textit{Re}}\gt 0.1$ ; blue – flow with $k\neq k_i$ and $\sigma _{{\textit{Nu}}}\leq 0.1$ , $\sigma _{\textit{Re}}\leq 0.1$ ; white and grey – chaotic flow; green – flow initialised with (3.2) is chaotic, but is steady with $k = k_i$ if initialised from a flow at a different ${\textit{Ra}}$ (see the end of the supplementary material available at https://doi.org/10.1017/jfm.2026.11293). The same symbol and colour coding is used in all figures of the paper.

3. Governing equations and DNS

The dimensionless Oberbeck–Boussinesq equations are obtained using the outer radius $R$ as length scale, the free-fall velocity $u_{{\textit{ff}}} \equiv ({\alpha \hat g R \varDelta })^{1/2}$ as velocity scale, the free-fall time $R/u_{{\textit{ff}}}$ as time scale, the temperature difference $\varDelta$ as temperature scale, and $\rho \alpha \hat g R \varDelta$ as pressure scale. The non-dimensional governing equations then read as follows: $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u} = 0$ ,

(3.1) \begin{align} D_{t}\boldsymbol{u} &= -\boldsymbol{\nabla }p + \sqrt {\frac {8\pi ^3}{\varGamma _\phi ^3}\frac {\textit{Pr}}{{{\textit{Ra}}}} } {\nabla} ^{2} \boldsymbol{u} + \frac {1}{\sqrt {1+{{\textit{Fr}}}^2}}\theta \boldsymbol{{e}}_{z} -\sqrt {{\varGamma _\phi \over 2\pi }} {\textit{Ro}} ^{-1} \boldsymbol{u} \times \boldsymbol{{e}}_{z} - \frac {{{\textit{Fr}}}}{\sqrt {1+{{\textit{Fr}}}^2}}\theta r \boldsymbol{{e}}_{r}, \nonumber \\ D_{t}\theta &= \sqrt {8\pi ^3 /( \varGamma _\phi ^3\,\textit{Pr}\,{{\textit{Ra}}})}\,{\nabla} ^{2}\theta , \qquad D_{t}\equiv \partial _t+({\boldsymbol{u}}\boldsymbol{\cdot }\boldsymbol{\nabla }), \end{align}

where $p$ is the pressure and $\boldsymbol{{e}}_{z}$ and $\boldsymbol{{e}}_{r}$ are the unit vectors in the vertical and radial directions, respectively. The boundary conditions are no-slip for the velocity at all walls, adiabatic at the top and bottom surfaces and isothermal at the vertical sidewalls ( $\theta _-$ and $\theta _+$ ).

To conduct DNS of CC, we use the latest version of the finite-volume code goldfish, which is fourth-order in space and third-order Runge–Kutta in time (Shishkina et al. Reference Shishkina, Horn, Wagner and Ching2015; Reiter et al. Reference Reiter, Zhang, Stepanov and Shishkina2021, Reference Reiter, Zhang and Shishkina2022). The code handles annular geometries (Yao et al. Reference Yao, Emran, Teimurazov and Shishkina2025), internal boundaries (Wagner & Shishkina Reference Wagner and Shishkina2015; Emran & Shishkina Reference Emran and Shishkina2020), specific boundary conditions (Ecke, Zhang & Shishkina Reference Ecke, Zhang and Shishkina2022; Zhang et al. Reference Zhang, Reiter, Shishkina and Ecke2024) and rotation with careful treatment of centrifugal effects (Zhang et al. Reference Zhang, van Gils, Horn, Wedi, Zwirner, Ahlers, Ecke, Weiss, Bodenschatz and Shishkina2020, Reference Zhang, Ecke and Shishkina2021).

The computational non-equidistant staggered grids with $N_r\times N_{\phi }\times N_z = 128\times 512\times 128$ for ${{\textit{Ra}}}\leq 2\times 10^6$ and $192\times 768\times 192$ for $2\times 10^6\lt {{\textit{Ra}}}\leq 10^7$ are sufficiently fine to resolve both the Kolmogorov and (for large $Pr$ more restrictive) Batchelor microscales. Within each thermal or viscous boundary layer, at least 10 grid points are used to satisfy DNS resolution criteria (Shishkina et al. Reference Shishkina, Stevens, Grossmann and Lohse2010). An additional grid refinement study for selected cases, with the grid resolution increased by a factor of at least 2.5, leads to changes in the Nusselt number ${\textit{Nu}}$ of less than $1\,\%$ over long-time averages.

To investigate multiple states in CC dominated by centrifugal buoyancy, we consider ${{\textit{Fr}}} = 10$ and $100$ and the range $2 \times 10^5 \leq {{\textit{Ra}}} \leq 10^7$ . In all our DNS, we fix $Pr = 4.3$ and the quantity $A$ by setting the outer annulus radius to $R = 6$ cm. For each set of control parameters, between 15 and 30 different initial conditions are tested (see figure 2). We initialise each flow with a zero velocity field, while the initial temperature field $\theta (t=0, r,\varphi ,z)=\theta _i(r,\varphi )$ mimics different numbers $k_i$ of roll pairs in the azimuthal direction:

(3.2) \begin{equation} \theta _i(r,\varphi ) = {\pi ^{-1}}\sin (k_i \varphi )\sin (2\pi r)+2r-{3}/{2}. \end{equation}

This function satisfies the boundary conditions $\theta _i|_{r=1/2}=-1/2$ and $\theta _i|_{r=1}=1/2$ , and is monotonic in the radial direction ( ${\partial \theta _i}/{\partial r} \geq 0$ ). The mid-height cross-sections of the initial flows and of the flows after 3000 time units are provided in the supplementary material available at https://doi.org/10.1017/jfm.2026.11293.

Figure 3.

( $a$ ) Time evolution of ${\textit{Re}}_{\varphi }/{\textit{Re}}_{r}$ and ${\textit{Nu}}$ for different initial roll-pair numbers $k_i$ at ${{\textit{Ra}}} = 5 \times 10^5$ and ${{\textit{Fr}}} = 10$ . Among different numbers of initial rolls $k_i$ , the persistent cases ( $k=k_i$ , shown with solid lines) are only for $3\leq k_i\leq 21$ . Non-persistent cases ( $k\neq k_i$ ) are shown with dashed lines. ( $b{-}d$ ) Instantaneous flow fields visualised with trajectories of passive tracer particles, coloured by the $z$ component of the vorticity, $\omega _z$ , for three representative cases.

4. The DNS results

Figure 2 summarises the initial and final states of all studied cases. Some flows retain the initial number of rolls (pink and orange), some become statistically steady but with a different final number of roll pairs $k$ than the initial $k_i$ (blue), some become chaotic (grey and white) and a subset of these chaotic cases can be brought to a statistically steady flow with a persistent roll structure when initialised from a steady flow at a different ${\textit{Ra}}$ with $k_i$ roll pairs (green). An example of the temporal flow development is shown in figure 3 for ${{\textit{Ra}}} = 5 \times 10^5$ and ${{\textit{Fr}}} = 10$ . Changes in the roll number from its initial value are reflected in abrupt variations of ${\textit{Nu}}$ and of the ratio between the Reynolds numbers based on the azimuthal and radial velocity components, ${\textit{Re}}_{\varphi }/{\textit{Re}}_r$ . Such changes can occur even after $10^3$ free-fall time units, indicating that the time required for a persistent coherent roll structure to develop does not scale with the convective time but rather with the diffusive time.

The final number of roll pairs $k$ obtained from different initial roll counts $k_i$ is shown in figure 4. Persistent cases lie along the diagonal, whereas non-persistent cases are scattered between $k=6$ and $14$ . Although the resulting $k$ values for non-persistent cases may appear random, they can be understood by examining the associated heat and momentum transport.

Figure 4.

Final number of roll pairs $k$ for different initial numbers of roll pairs $k_i$ at ${{\textit{Fr}}} = 10$ ( $a$ ) and ${{\textit{Fr}}} = 100$ ( $b$ ). Symbol meanings are the same as in figure 2.

The results for the heat and momentum transport, expressed by ${\textit{Nu}}$ and ${\textit{Re}}$ as functions of the initial number of roll pairs $k_i$ , are shown in figure 5 and can be summarised as follows. For fixed ${\textit{Fr}}$ , persistent statistically steady flows (coloured symbols) occur only for small and moderate ${\textit{Ra}}$ , since at larger ${\textit{Ra}}$ the flows become chaotic (white and grey symbols). For moderate ${\textit{Ra}}$ , both ${\textit{Nu}}$ and ${\textit{Re}}$ increase with growing $k_i$ (i.e. decreasing aspect ratio of rolls, $\varGamma _r$ ), in agreement with previous findings for Rayleigh–Bénard convection (Wang et al. Reference Wang, Verzicco, Lohse and Shishkina2020); see the pink pentagons in figure 5(a,c) for ${{\textit{Ra}}}=2\times 10^6$ , ${{\textit{Fr}}}=10$ , and down-pointing triangles in figure 5(b,d) for ${{\textit{Ra}}}=5\times 10^6$ , ${{\textit{Fr}}}=100$ . With a further increase of $k_i$ , the flow becomes chaotic and both ${\textit{Nu}}$ and ${\textit{Re}}$ drop to lower values (grey symbols). For small ${\textit{Ra}}$ , both ${\textit{Nu}}$ and ${\textit{Re}}$ likewise first increase with $k_i$ , reach a maximum and then start to decrease owing to full laminarisation of the flow (see the remaining pink symbols in figure 5). In this case, very narrow rolls (very large $k$ ) are damped by viscosity, dissipation rises, the heat transport becomes more diffusive and the flow approaches a steady (or weakly unsteady) laminar state. For sufficiently large $k_i$ , the flow can no longer sustain this high roll number and undergoes an abrupt structural change to a steady flow with a smaller number of rolls $k\lt k_i$ (blue symbols). During this transition, ${\textit{Nu}}$ and ${\textit{Re}}$ jump to values close to their maxima and thereafter remain almost independent of further increases in $k_i$ .

The ranges of possible ${\textit{Nu}}$ and ${\textit{Re}}$ variations with $k_i$ , as obtained in our DNS, are listed in table 1. The largest variations in ${\textit{Nu}}$ and ${\textit{Re}}$ for different $k_i$ occur at smaller ${\textit{Ra}}$ (up to ${\approx} 67\,\%$ in ${\textit{Nu}}$ for ${{\textit{Ra}}} = 2\times 10^5$ ). At our largest ${{\textit{Ra}}}=10^7$ , they are much smaller ( ${\lt} 4\,\%$ in ${\textit{Nu}}$ ) but remain finite. Changes in ${\textit{Nu}}$ are always larger than those in ${\textit{Re}}$ .

Table 1.

Comparison of the maximum ( ${\textit{Nu}}_{\textit{max}}$ , ${\textit{Re}}_{\textit{max}}$ ) and minimum ( ${\textit{Nu}}_{\textit{min}}$ , ${\textit{Re}}_{\textit{min}}$ ) values of the Nusselt and Reynolds numbers for all cases at the same control parameters but different initial conditions.

Figure 5.

Nusselt numbers ( $a,b$ ) and Reynolds numbers ( $c,d$ ) for different initial numbers of roll pairs $k_i$ at ${{\textit{Fr}}} = 10$ ( $a,c$ ) and ${{\textit{Fr}}} = 100$ ( $b,d$ ). Symbol meanings are the same as in figure 2.

5. Estimation of the feasible number of azimuthal rolls

In figure 6(a,b) we show the number of roll pairs $k$ in CC flows with coherent roll structures as functions of ${\textit{Ra}}$ . Flows in which the initial and final roll numbers coincide ( $k = k_i$ ) are marked in pink and orange, while those with $k \neq k_i$ are marked in blue. Green symbols denote flows that are chaotic when initialised with (3.2) and are steady when initialised with a flow with $k$ roll pairs at a different ${\textit{Ra}}$ . For fixed ${\textit{Fr}}$ , the range of possible states is broad at small ${\textit{Ra}}$ and shrinks as ${\textit{Ra}}$ increases. How can we estimate these ranges?

5.1. Estimate of the maximal number of roll pairs from Poincaré–Friedrichs inequality

According to the Poincaré–Friedrichs inequality, for a dimensionless velocity $\boldsymbol{u}$ that vanishes on the boundary, one has $\hat \lambda _{\textit{min}}\langle {\boldsymbol{u}}^2\rangle \leq \langle (\boldsymbol{\nabla }{\boldsymbol{u}})^2\rangle$ , where $\hat \lambda _{\textit{min}}$ is the smallest positive eigenvalue of the Laplacian, associated with an eigenfunction $f$ that has a non-zero projection on at least one component of the velocity $u$ (i.e. $\langle uf\rangle \neq 0$ ), and $\lambda _{\textit{min}}=L^2\hat \lambda _{\textit{min}}$ its dimensionless analogue. Introducing the integral quantity

(5.1) \begin{equation} B\equiv {\textit{Re}}^{2}/\left(L^{4}\nu ^{-3}\epsilon _u\right)\!, \end{equation}

where $\epsilon _u\equiv \nu \langle (\boldsymbol{\nabla }{\boldsymbol{u}})^2\rangle$ is the averaged kinetic energy dissipation rate, we obtain $\lambda _{\textit{min}} B\leq 1$ .

Now, to estimate how many azimuthal rolls can fit inside the annulus, we consider the Laplace eigenvalue problem with zero boundary conditions on the solid walls, and $2\pi$ -periodicity in the $\varphi$ direction. We seek eigenfunctions of $(-{\nabla} ^2)$ that exhibit a prescribed number of oscillations in the azimuthal direction. Specifically, we consider the eigenvalue problem $-{\nabla} ^2 f = \lambda f$ in $\varOmega$ , subject to the appropriate boundary conditions. The Laplacian is then expressed in cylindrical coordinates, $ {\nabla} ^2 f = {r^{-1}}{\partial _r}\! (r{\partial _r f} ) + {r^{-2}}{\partial _{\varphi }^2 f} + {\partial _z^2 f}$ . The eigenfunctions separate in $(r,\varphi ,z)$ , $f = P(r)\,\varPhi (\varphi )\,Z(z)$ , so that the vertical problem with zero conditions at $z=0,H/L$ gives $Z_n(z) = \sin ({n\pi zL/H} )$ , $n\geq 1$ , and the azimuthal problem with $2\pi$ -periodicity yields $\varPhi _k(\varphi ) = \cos (k\varphi )$ or $\sin (k\varphi )$ . Such a function undergoes exactly $2k$ sign changes when $\varphi$ runs from $0$ to $2\pi$ ; hence $k$ corresponds directly to the number of roll pairs. Introducing $ \varkappa ^2 \equiv \lambda - (n\pi L/H)^2$ , we obtain the radial equation $ r^2 P'' + r P' + (\varkappa ^2 r^2 - k^2)\,P = 0$ , whose solutions are $ P(r) = a J_k(\varkappa r) + b Y_k(\varkappa r)$ , with $J_k$ and $Y_k$ being the Bessel functions of the first and second kind. The Dirichlet boundary conditions at $r=R/L-1$ and $r=R/L$ impose $ a J_k(\varkappa (R/L-1)) + b Y_k(\varkappa (R/L-1)) = 0$ and $ a J_k(\varkappa R/L) + b Y_k(\varkappa R/L) = 0$ , for which non-trivial solutions for $a$ and $b$ exist only if

(5.2) \begin{equation} F_k(\varkappa ) \equiv J_k(\varkappa (R/L-1))\,Y_k(\varkappa R/L) - J_k(\varkappa R/L)\,Y_k(\varkappa (R/L-1)) = 0. \end{equation}

For each triple $(k,j,n)$ , the eigenvalue equals $\lambda _{k,j,n} = \varkappa _{k,j}^2 + ({n\pi L/H} )^2$ , where $\varkappa _{k,j}$ is the $j$ th positive root of $F_k(\varkappa )=0$ . The smallest eigenvalue for a given $k$ is obtained for $j=n=1$ . For the annulus considered in our DNS, the radial condition (5.2) simplifies to

(5.3) \begin{equation} F_k(\varkappa )= J_k(\varkappa )\,Y_k(2\varkappa ) - J_k(2\varkappa )\,Y_k(\varkappa ) = 0. \end{equation}

The lowest eigenvalue for the wavenumber $k$ and the corresponding eigenfunction are

(5.4) \begin{equation} {\lambda _{\textit{min}}(k)} = \varkappa _{k,1}^2 + \pi ^2,\quad \text{where }\varkappa _{k,1} \text{ is the first positive root of (5.3)}, \end{equation}

and $f_k=[J_k(\varkappa _{k,1} r) Y_k(2\varkappa _{k,1}) - Y_k(\varkappa _{k,1} r) J_k(2\varkappa _{k,1})]\sin (k\varphi )\sin (\pi z)$ , respectively.

Figure 6.

Number of roll pairs $k$ in CC flows with coherent roll structures after 3000 time units, as functions of ${\textit{Ra}}$ ( $a,b$ ) and $B$ ( $c,d$ ) for ${{\textit{Fr}}} = 10$ ( $a,c$ ) and ${{\textit{Fr}}} = 100$ ( $b,d$ ). Symbols, DNS results; lines, theoretical upper bounds from the Poincaré–Friedrichs inequality (magenta) and upper and lower bounds (5.7) from the elliptical stability criteria (red and yellow). Symbol meanings are the same as in figure 2, with most blue symbols overlapping the pink ones; chaotic cases are not shown.

By solving the inverse problem numerically, one can determine which values of $k$ are admissible for a prescribed value of $B$ . More precisely, for each $k$ the eigenvalue $\lambda _{\textit{min}}(k)$ is obtained from (5.4), which directly yields the maximal value of $B$ , $B_{\textit{max}} = {\lambda _{\textit{min}}^{-1}(k)}$ , for which a flow state with $2k$ azimuthal rolls can exist. From this, one can readily evaluate the maximal number of roll pairs $k_{\textit{max}}$ that can develop for a given $B$ . In figure 6(c,d) this theoretical estimate is shown by the magenta lines and exhibits very good agreement with the DNS data. The dependence of $k_{\textit{max}}$ on $B$ can be directly translated into a dependence on ${\textit{Ra}}$ . Since $k_{\textit{max}}(B)$ is a monotonically decreasing function, for each ${\textit{Ra}}$ we take the smallest value of $B$ obtained in the DNS at that ${\textit{Ra}}$ and thus obtain the corresponding dependence $k_{\textit{max}}({{\textit{Ra}}})$ for each ${\textit{Fr}}$ . These dependences are plotted in figure 6(a,b) as magenta lines; they show good agreement with the DNS data and provide reliable estimates of the maximal possible number of roll pairs $k$ in statistically steady flows with coherent roll structures, using only the minimal value of the integral quantity $B$ for given ${\textit{Ra}}$ and ${\textit{Fr}}$ .

5.2. Bounds on the number of roll pairs from elliptical instability

The three-dimensional destabilisation of vortices with stretched elliptical streamlines of the aspect ratio $\varGamma _r\gt 1$ is usually attributed to the elliptical instability (Bayly Reference Bayly1986; Waleffe Reference Waleffe1990; Zwirner et al. Reference Zwirner, Tilgner and Shishkina2020). For a columnar vortex with angular velocity $\omega$ and ellipticity $\beta \equiv ({\varGamma _r^2 - 1})/({\varGamma _r^2 + 1})$ , inviscid asymptotics give a maximum growth rate $\sigma$ of order $\beta \omega$ , while viscosity introduces a damping of order $E^{1/2}\omega$ with the vortex Ekman number

(5.5) \begin{equation} E\equiv \nu /\bigl(\omega D^2\bigr)\approx (L/D)^2\sqrt {B}/\textit{Re}, \quad D^2\equiv L^2{\left(\varGamma _r^2+1\right)/2}, \end{equation}

leading to $ \sigma \approx (c_1\,\beta - c_2\,E^{1/2})\omega$ with $c_1\gt 0$ and $c_2\gt 0$ (Hollerbach & Kerswell Reference Hollerbach and Kerswell1995; Lacaze, Gal & Dizès Reference Lacaze, Gal and Dizès2005). Thus the stretched roll flow can sustain only if $\beta \lt (c_2/c_1) E^{1/2}$ , i.e.

(5.6) \begin{equation} \left({\varGamma _r^2 - 1}\right)^2/\left({\varGamma _r^2 + 1}\right)\lt 2\left(c_2/c_1\right)^2\sqrt {B}/\textit{Re}\equiv s. \end{equation}

Estimating $\varGamma _r \approx 3\pi /(2k)$ , from relation (5.6) we obtain the lower and upper bounds for $k$ :

(5.7) \begin{equation} 3\pi \left({4+2s+2\bigl({s^2+8s}\bigr)^{1/2}}\right)^{-1/2}\leq k\leq 3\pi \left({4+2s-2\bigl({s^2+8s}\bigr)^{1/2}}\right)^{-1/2}. \end{equation}

Our DNS data (albeit for only two values of ${\textit{Fr}}$ ) suggest that $s \approx 10^5 {{\textit{Fr}}}^{0.3}{{\textit{Ra}}}^{-0.8}$ , showing a clear decrease of $s$ with increasing ${\textit{Ra}}$ . Since the range (5.7) of admissible $k$ values around $k=3\pi /2$ (corresponding to the roll aspect ratio $\varGamma _r=1$ ) shrinks as $s$ decreases, we conclude that, for any fixed ${{\textit{Fr}}}\gg 1$ , the range of possible statistically steady states with coherent roll structures narrows with increasing ${\textit{Ra}}$ (see figure 6, red and yellow lines).