2. Overview

For each pair of control parameters $(\textit{Ra},\textit{Fr})$ , the simulations are started from temperature fields seeded with a prescribed initial number of azimuthal roll pairs (as shown in figure 1 b–d), $k_i$ . The system is then evolved for long times to determine the final roll pair number $k$ , and whether the state is persistent ( $k=k_i$ ), reorganises ( $k\neq k_i$ ) or becomes chaotic. In total, the authors simulated 278 cases and the main message is that CC exhibits pronounced flow-memory; for a wide range of parameters, distinct roll states are dynamically possible at the same $(Ra,Fr)$ , as shown in figure 1. For example, for ${\textit{Ra}} = 2\times 10^5$ and ${\textit{Fr}}=10$ , the flow could be in 15 different states, with the number of convection rolls ranging from 2 pairs to 16 pairs. In some regimes, coherent roll states can only be reached if the flow is initialised from a statistically steady state at a different ${\textit{Ra}}$ , highlighting strong history dependence.

An important consequential aspect of the multiple states is what it does to the global response. In CC, the number of roll pairs controls the characteristic roll size and thus the efficiency of radial mixing and transport of heat and momentum. Yao et al. (Reference Yao, Emran, Teimurazov, Kriening, Song and Shishkina2026) show that, at fixed $(Ra,Fr)$ , the Nusselt and Reynolds numbers can differ dramatically between coexisting states in weakly to moderately turbulent regimes (the variations of $Nu$ and $Re$ can be up to $66.8\,\%$ and $57.9\,\%$ , respectively), with the largest transport typically associated with intermediate roll numbers. When the imposed roll number is too small or too large, the system either reorganises to a different roll count or becomes chaotic. A further trend is equally important for general turbulence, namely that the availability of distinct coherent states is not constant. As the ${\textit{Ra}}$ number increases, the range of multiple states narrows. At sufficiently large ${\textit{Ra}}$ , only chaotic dynamics remain and the dependence of transport on the initial number of rolls becomes weak. In other words, intensifying turbulence progressively erodes the robustness of coherent roll organisations.

Figure 1.

(a) Number of roll pairs $k$ in centrifugal convection when the flow becomes statistically steady (after 3000 free fall time), as functions of ${\textit{Ra}}$ for ${\textit{Fr}}$ = 10. Different symbol shapes indicate different ${\textit{Ra}}$ . Symbols represent DNS results: pink and yellow represent flow with a persistent coherent roll structure, where initial ( $k_i$ ) and final ( $k$ ) roll numbers coincide, $k_i=k$ ; blue represents flow with $k_i\neq k$ (most blue symbols overlapping the pink ones); green represents flow ends up chaotic. Lines represent theoretical upper bounds from the Friedrichs inequality (magenta), and upper and lower bounds from the elliptical stability criteria (red and yellow). (b–d) Three examples of the total 19 flow states for ${\textit{Ra}}=5\times 10^5$ and ${\textit{Fr}}=10$ , with 21, 6 and 4 pairs of convection rolls, respectively.

A strength of Yao et al. (Reference Yao, Emran, Teimurazov, Kriening, Song and Shishkina2026) is that it moves beyond classification and gives two complementary, physically intuitive constraints on which roll states can exist. First, the authors derive an upper limit on the number of roll pairs from a global mathematical constraint based on the Poincaré–Friedrichs inequality, which states that in a confined domain, the kinetic energy of a flow is bound by the magnitude of its velocity gradients. In the annulus considered here, once a specific number of roll pairs is prescribed their azimuthal wavelength is fixed by annulus geometry. Fitting more rolls into the same circumference makes each roll narrower, and therefore forces the velocity field to vary more rapidly in space. More rapid variation implies stronger gradients and thus greater viscous dissipation. Through the corresponding Laplacian eigenvalue for that azimuthal wavenumber, this geometric requirement becomes a quantitative lower bound on the dissipation associated with that roll structure. Because this lower bound strengthens as the roll number increases, only small $k$ remains compatible with the global constraint. The resulting theoretical bound on the maximal admissible roll number agrees closely with the DNS state diagram. Second, Yao et al. (Reference Yao, Emran, Teimurazov, Kriening, Song and Shishkina2026) invoke an elliptical-instability mechanism to explain why the admissible range of flow state numbers contracts as ${\textit{Ra}}$ increases. Coherent rolls can be viewed as vortical structures with strained, elliptically deformed streamlines. Such vortices are known to be susceptible to three-dimensional destabilisation when the strain is sufficiently strong relative to viscous damping (Zwirner et al. Reference Zwirner, Tilgner and Shishkina2020). By estimating the instability growth rate and balancing it against viscous effects, the authors derive both lower and upper bounds on the admissible roll numbers. Rolls that are too extended or too compressed are therefore more vulnerable to three-dimensional breakdown. The upper and lower bounds found by Yao et al. (Reference Yao, Emran, Teimurazov, Kriening, Song and Shishkina2026) agree very well with their DNS results, as shown in figure 1.

3. Future

The study of the multiple states in centrifugal convection reinforces the view that multiple turbulent states are not special exceptions, but a generic possibility whenever turbulence coexists with large-scale organisation. Several open directions follow. If transport can differ by order-one factors between coexisting states, then ‘scatter’ in $Nu/Re$ or drag laws may reflect mixed states rather than experimental uncertainty. A practical next step is to develop state-conditioned scalings, transport laws that explicitly condition on the large-scale organisation (here, roll number), as was done in Rayleigh–Bénard convection previously (Xi & Xia Reference Xi and Xia2008; Weiss & Ahlers Reference Weiss and Ahlers2011; van der Poel et al. Reference van der Poel, Stevens and Lohse2011, Reference van der Poel, Stevens, Sugiyama and Lohse2012).

The strong history dependence highlighted by Yao et al. (Reference Yao, Emran, Teimurazov, Kriening, Song and Shishkina2026) suggests both a challenge and an opportunity. The challenge is reproducibility, where the protocol becomes part of the ‘definition’ of the state. The opportunity is control; if weak perturbations can select among states with different transport, one might tune heat or momentum transfer by deliberately steering the flow into a desired basin. Our recent experimental study of multiple states in large aspect ratio (ratio between width and height of the convection cell is 10) Rayleigh–Bénard convection is such an attempt (Li et al. Reference Li, Huo, Chen and Xi2026). Finally, it is worth asking how widely this combination of global energetic constraints (Poincaré–Friedrichs inequality) and local instability (elliptical instability) mechanisms applies. Similar logic may help rationalise multistability in other turbulent flows with dominant coherent structures, from wall-bounded rolls and streaks to large-scale circulation patterns in convection and rotating turbulence. Therefore, the contribution of Yao et al. (Reference Yao, Emran, Teimurazov, Kriening, Song and Shishkina2026) is broader than for centrifugal convection only. It offers a clean demonstration that multiple states in turbulence can be systematically mapped, that their transport consequences can be substantial and that their existence can be constrained by simple physical principles.