Flow reorganisation and intermittent cessations of the large-scale convection (LSC) are investigated in turbulent liquid-metal Rayleigh–Bénard convection at a Prandtl number $ \textit{Pr} \thickapprox 0.03$. Flow measurements are performed at Rayleigh numbers of $\textit {Ra} = 9.33 \times 10^6, 5.31 \times 10^7$ and $6.02 \times 10^8$ in a cylindrical sample of aspect ratio $\varGamma$ = 0.5 filled with the ternary alloy GaInSn. The LSC structure is characterised using two complementary diagnostics: the classical multi-thermoprobe method based on sidewall temperature measurements and contactless inductive flow tomography (CIFT), which reconstructs the three-dimensional velocity field and enables a proper orthogonal decomposition of the dominant flow modes. While both diagnostics provide consistent descriptions of well-organised circulation states, they increasingly diverge during strongly three-dimensional flow reorganisation as they probe different physical quantities. The volumetric CIFT reconstructions reveal that cessations are associated with a temporary loss of global flow coherence accompanied by a redistribution of kinetic energy from the dominant single-roll and double-roll circulation modes towards higher-order structures. In contrast, localised flow reorganisations are identified as events representing transitions between competing circulation modes without complete loss of global coherence. The transition pathways are highly stochastic and show no unique modal sequence. The results provide the first structurally resolved experimental characterisation of the dynamics of flow cessations in turbulent liquid-metal convection and demonstrate the role of flow mode transitions in triggering large-scale intermittency at very low Prandtl numbers.

The spectral budget of the vertical heat flux $\langle w\theta \rangle$ is investigated in turbulent Rayleigh–Bénard convection by direct numerical simulation. Eight cases spanning three Rayleigh numbers ($\textit{Ra} = 1\times 10^5$, $6\times 10^6$ and $4\times 10^8$) and up to three geometric aspect ratios each are computed at $\textit{Pr} = 1$. The premultiplied spectra of $\langle w\theta \rangle$ reveal a two-peak structure when both $\textit{Ra}$ and $\varGamma$ are sufficiently large. Production of $\langle w\theta \rangle$ is dominated by temperature fluctuations; production by velocity fluctuations is confined to the near-wall region and the large-scale production that develops at sufficient $\varGamma$ is restricted to the bulk. At large $\varGamma$ an inverse inter-scale cascade transfers $\langle w\theta \rangle$ from intermediate to superstructure scales, the transfer towards the superstructure scale taking place away from the boundary layer. This upscale transfer is specific to the covariance, as the vertical kinetic energy and the temperature variance cascade towards small scales, so that the turbulence selectively transfers the correlation between vertical velocity and temperature towards the superstructure scale, consistent with the strong large-scale coherence of the superstructures. Pressure transport is the only mechanism that delivers large-scale $\langle w\theta \rangle$ towards the wall with appreciable magnitude, yet the pressure–strain term destroys the transported quantity so that the net pressure–temperature interaction is purely negative at all scales. No mechanism in the budget delivers large-scale spectral content into the boundary layer, providing a scale-by-scale explanation for the insensitivity of the Nusselt number to the aspect ratio beyond a moderate threshold.

This study investigates how the impact of an internal heat source and anisotropic thermal diffusivity influences convective motion in a horizontally inclined porous medium saturated with fluid, assuming an adiabatic bottom boundary and an isothermal top boundary. Here, the momentum equation is formulated using the Darcy–Forchheimer model and assumes that the viscosity is a linear function of the temperature. Linear and nonlinear stability analyses have been conducted to determine the stability by calculating the critical Rayleigh number and the associated wavenumber. For nonlinear stability, conditional stability has been established for all initial data with higher-order norms, showing that the $L^2$ theory is insufficient to achieve similar stability, where $L^2$  represents Hilbert space over the domain  $\omega$. It has also been shown that the conditional stability bound can be attained by restricting the internal heat source parameter Q using higher-order norms. The Chebyshev tau-QZ algorithm is used to solve the generalised eigenvalue problem, which is obtained by employing the normal mode technique on the disturbance equations. The effect of various dimensionless parameters, such as anisotropic thermal diffusivity parameters $(\xi )$, Rayleigh number (Ra), angle of inclination ($\phi$) and viscosity parameter $(\varGamma )$, on the stability of the medium is investigated, and streamlines are also shown by contour plots to depict the flow pattern. The results show that the system is stable when $\xi \gt 1$ and unstable when $\xi \lt 1$, favouring convection. It is also observed that the convection is delayed with rising values of $\phi$, and the region of subcritical instability increases uniformly with rising values of $\phi$.

We present a comprehensive experimental investigation on flow mode transition and heat transport in confined liquid metal convection at a Prandtl number of $ \textit{Pr} = 0.029$, using cylindrical convection cells with aspect ratios $\varGamma = 1$, $1/2$ and $1/3$. Whenever comparable data are available, the present low-$ \textit{Pr}$ results are compared with previous measurements in moderate-$ \textit{Pr}$ water convection (Xi & Xia, Phys. Fluids vol. 20, 2008, p. 055104; Weiss & Ahlers, J. Fluid Mech. vol. 676, 2011, pp. 5–40). It is found that the single-roll mode (SRM) dominates in the $\varGamma = 1$ and $1/2$ cells, whereas the double-roll mode (DRM) prevails in the $\varGamma = 1/3$ cell. The relative weight of the SRM increases with $\varGamma$, while those of the DRM and the transition mode decrease. Focusing on the $\varGamma = 1/3$ cell, three representative DRM–SRM–DRM transition routes are observed: one induced by cessations and two associated with azimuthal rotational reorganisation of the flow. Beyond the previously studied reversals of the SRM, the DRM can also reverse through either cessation or continuous azimuthal reorientation. In this strongly confined cell, the dominant DRM is asymmetric, consisting of a larger and stronger upper roll, and a smaller and weaker lower roll. These changes in flow organisation are also reflected in heat transport: the SRM transports heat slightly more efficiently than the DRM and the difference decreases with increasing Rayleigh number $ \textit{Ra}$. For $\varGamma = 1/2$, comparison with moderate-$ \textit{Pr}$ water convection further suggests that this heat-transport difference depends weakly on $ \textit{Pr}$, compared with its stronger dependence on $ \textit{Ra}$. These results deepen the understanding of large-scale flow structures in turbulent thermal convection and provide useful data for modelling large-scale flows in turbulent convection.

We investigate the propagation of counter-rotating convection rolls in Rayleigh–Bénard convection initiated locally in a quiescent fluid layer under supercritical conditions. The velocity of the front separating quiescent fluid from the forming convection rolls, and the wavenumber of the convection rolls remaining behind the front, are explored. We numerically investigate fronts of forming convection rolls over five orders of magnitude of the reduced Rayleigh number, $\epsilon$, in two- and three-dimensional domains, for a broad range of boundary conditions, and for different front initiation approaches. In all cases, the front velocity increases as $\epsilon ^{1/2}$ with increasing $\epsilon$ for $\epsilon \lesssim 1$ in agreement with predictions using the amplitude equation. The amplitude equation description of the front velocity remains accurate for $\epsilon \lesssim 10$ except when the Prandtl number is large which yields a velocity that is faster than predicted for a fluid layer far from threshold. The wavenumber of the convection rolls increases linearly with $\epsilon$ in agreement with the wavenumber that maximises the growth rate of perturbations in the linear regime. Farther from onset, the wavenumber growth transitions to a reduced scaling of $\epsilon ^{1/4}$ in agreement with predictions using the Swift–Hohenberg equation in the large-$\epsilon$ limit. The scalings describing the wavenumber variation with $\epsilon$ are independent of the domain geometry, boundary conditions and front initiation method. However, the front-selected wavenumber at criticality does not equal the critical wavenumber of the bulk instability, in general, and depends significantly upon these details. We compare our results with experimental measurements where possible.

This study examines anomalous thermal transport in confined liquid metal Rayleigh–Bénard convection via direct numerical simulations over Rayleigh numbers $\textit{Ra} \in [10^5, 10^8]$. A phenomenological hybrid model is introduced to capture the full transition from the near-wall boundary layer to the non-monotonic core, and the presence of an inverse mean temperature gradient in the bulk is shown to originate from the spatial organization of dynamic thermal blobs. For temperature fluctuations, a spatial characterization of the temperature standard-deviation profile ($\sigma _T$) reveals a distinctive bimodal structure, with a secondary peak emerging away from the wall. Through variation of spanwise confinement, no-slip sidewalls are found to constrain the development of corner vortices, reorganizing them into a coherent spiral-like precession. This three-dimensional flow topology induces intense spanwise velocity fluctuations, which are associated with the transport and trapping of boundary-layer thermal plumes in the core region, thereby supporting the formation of the anomalous near-core $\sigma _T$ peak. At the largest Rayleigh numbers considered, the strengthening of turbulent mixing and large-scale circulation weakens this coherent corner-vortex activity, leading to the attenuation of the bimodal signature. Furthermore, we show that such quasi-two-dimensional confinement modifies the macroscopic heat transfer scaling, producing a steeper effective exponent ($\textit{Nu} \sim \textit{Ra}^{0.332}$) compared with classical predictions – a phenomenon closely tied to the morphological condensation of thermal plumes.

We study by means of direct numerical simulations the influence of the dimensionality of convection on flow properties. We call attention to a few general principles from considering in totality the results from one dimension, two dimensions, three dimensions and four dimensions. In particular, we explore two practical aspects: (i) the transient time, or the amount of time it takes for the flow to reach the steady state; and (ii) possible implications for the so-called ultimate state.

We perform two-dimensional direct numerical simulations of rapidly rotating annular centrifugal convection to investigate how mixed (asymmetric) velocity boundary conditions and geometric curvature shape the flow organisation and heat transfer in the quasi-two-dimensional, rotation-constrained regime. Motivated by the quasi-two-dimensionalisation under strong rotation and the long spin-up required for large-scale states, we employ two-dimensional simulations and consider four boundary-condition sets: no slip/no slip (INON), no slip/stress free (INOS), stress free/no slip (ISON) and stress free/stress free (ISOS). For fixed geometry with the radius ratio $\eta =0.5$ and over the Rayleigh-number range $ \textit{Ra}\in [10^6,10^9]$, the heat transfer is strongest for ISOS, followed by INOS and INON, while ISON exhibits a pronounced suppression as a strong zonal flow aligned with the rotation develops. In the three cases where the flow remains in a roll-dominated state, the Nusselt number $ \textit{Nu}$ follows an effective classical-type scaling close to $ \textit{Nu}\sim Ra^{0.27}$, whereas the zonal-flow branch displays a much weaker scaling $ \textit{Nu}\sim Ra^{0.1}$ and strong flow anisotropy with a large difference between the radial and azimuthal Reynolds numbers $ \textit{Re}_r\ll Re_\varphi$. A dissipation analysis shows that zonal-flow formation is accompanied by a transition from boundary-layer-dominated dissipation to a relatively low and uniform bulk dissipation, consistent with shear-induced plume suppression. Using a representative curvature scan at fixed $ \textit{Ra}=10^7$, we find that, within this quasi-two-dimensional, strongly rotating regime, increasing $\eta$ weakens the curvature asymmetry and makes the ISON zonal-flow branch less robust, with the flow approaching roll-dominated convection in the planar limit; the accompanying bulk-temperature asymmetry is interpreted in terms of the boundary heat-flux asymmetry using a free-convective boundary-layer model.

A weakly nonlinear analysis is used to investigate the combined effects of viscous dissipation and inclination on Darcy–Bénard convection. For a horizontal layer and in the absence of viscous dissipation then, as the Darcy–Rayleigh number is increased, the first pattern to arise takes the form of two-dimensional rolls. If a weak viscous dissipation effect is included for a horizontal layer, this causes the preferred pattern near onset to change to a hexagonal planform. Here, we consider the additional effect of introducing an inclination to the layer. This causes an increase in the critical Darcy–Rayleigh number for each of the constituent rolls of the hexagon; this size of this increase depends on the orientation of the roll. In consequence, the hexagons no longer consist of three rolls of equal amplitudes and this modifies the stability criteria substantially. A detailed analysis is given for the range of existence and stability of both longitudinal rolls and what are termed pseudo-hexagons. It is shown that there is a maximum inclination for which a stable three-dimensional convection pattern is possible.

We investigate the heat transport in porous media convection (PMC) over a wide Rayleigh–Darcy number range of $26.8\leqslant Ra\leqslant 2.62\times 10^5$, and a Darcy number range of $6.18\times 10^{-7}\leqslant Da\leqslant 1.21\times 10^{-5}$. In the experiments, we employ 3D-printed lattice structures as the solid porous matrix and water as the working fluid. Quantitative analyses of the porous medium Nusselt number $ \textit{Nu}_m$ and local temperature statistics reveal that the present system undergoes a transition through five distinct regimes: (I) conduction, (II) convection, (III) oscillation, (IV) transition, (V) classical Rayleigh–Bénard convection. This transitional process bridges the gap between Rayleigh–Darcy-like behaviour and Rayleigh–Bénard-like behaviour in PMC. By varying the permeability of the matrix, we further examine the role of the Darcy number $ \textit{Da}$, which turns out to have a profound impact on the transitional processes across different regimes. Flow field measurements reveal that the flow structures within regime IV and regime V evolve from several horizontally stacked convection rolls to a single-roll structure, and the pore-scale Reynolds numbers both exceed unity in these two regimes. Finally, we report the corresponding phase diagram in the $ \textit{Ra}$–$ \textit{Da}$ space.

We study the navigation of a self-propelled inertial particle in two-dimensional Rayleigh–Bénard convection at Prandtl number $\textit{Pr}=0.71$ and cell aspect ratio $\varGamma =4$ for Rayleigh numbers $Ra$ ranging from $10^7$ to $10^{11}$. A reinforcement-learning (RL) controller selects the propulsive acceleration, subject to an upper bound $\mathcal{A}_{\textit{max}}$, to achieve a prescribed horizontal displacement. We find that the success rate increases abruptly with $\mathcal{A}_{\textit{max}}$ at moderate $Ra$, whereas at higher $Ra$ the transition becomes more gradual and shifts to larger $\mathcal{A}_{\textit{max}}$. Moreover, although the completion time increases with $Ra$, the propulsion energy required for successful traversal decreases. Proper orthogonal decomposition indicates that these performance differences are associated with reorganisation of the carrier flow. At moderate $Ra$, the dominant large-scale circulation partitions the domain through persistent transport barriers, requiring a finite thrust surplus to cross them; at higher $Ra$, energy is distributed across many modes, the barriers fragment and transient plume-assisted pathways emerge. Compared with a constant-heading baseline, the learned policy aligns with local currents and consumes significantly less energy. Lagrangian coherent structure analysis further suggests that the RL agent tends to cross repelling barriers and surf along attracting pathways. Finally, by mapping these behaviours onto the local Eulerian flow topology using Voronoi tessellation and the $Q$-criterion, we distil an interpretable, physics-based heuristic strategy that retains robust navigability. These results connect turbulent-flow organisation with autonomous navigation under bounded actuation.

Turbulent convection is a fundamental transport process that shapes weather and climate, powers flows in planetary interiors and stars, and limits the performance of energy and heat-transfer technologies. Yet how these flows organise simultaneously into large-scale ‘superstructures’ and small-scale near-wall patterns remains unclear. Here, using Rayleigh–Bénard convection with large domain size, we show that both structures are captured by optimal linear modes that maximise transient energy amplification, identified via linear analysis based on turbulent mean profiles. Two amplification regimes emerge with clear scale separation: large-scale modes spanning the full central domain with horizontal wavelengths ${\sim} 6$ times the plate separation and small-scale modes confined near the walls at ${\sim} 11$ times the thermal boundary layer thickness. These results indicate that linear energy amplification plays an important organising role in multiscale turbulent convection and establish a unifying link to analogous non-modal processes in wall-bounded shear turbulence.

We report an experimental investigation of turbulent Rayleigh–Bénard convection in a quasi-two-dimensional rectangular cell of large aspect ratio ($\varGamma = 10$) over the Rayleigh number range $5.4\times 10^7 \leqslant Ra \leqslant 7.2\times 10^9$ and Prandtl number range $4.3 \leqslant \textit{Pr} \leqslant 67.3$. Planar particle image velocimetry measurements show that the flow self-organises into several horizontally aligned convection rolls, and repeated experiments under identical parameters (both $Ra$ and $\textit{Pr}$) reveal that the number of rolls varies within the range of 3–7 with 6 being the most probable, which demonstrates the presence of multiple flow states. When $\textit{Pr}$ is increased to 67.3, the number of roll-like structures increases significantly, indicating a structural transition from a roll-dominated to a plume-dominated flow. This transition is reflected in the global momentum transport, for $\textit{Pr} \leqslant 18.3$, the Reynolds number scales as $\textit{Re} \sim Ra^{0.58}\textit{Pr}^{-0.97}$, whereas the scaling is changed to $\textit{Re} \sim Ra^{0.72}$ when $\textit{Pr}$ reaches 67.3. Within individual rolls, we further examine the Reynolds numbers based on horizontal and vertical velocity components, $\textit{Re}_{u,\textit{roll}}$ and $\textit{Re}_{w,\textit{roll}}$, and find that the former increases while the latter decreases with roll size (quantified as the aspect ratio of the roll $\varGamma _{\textit{roll}}$) due to continuity constraints, with their ratio following $\textit{Re}_{w,\textit{roll}}/\textit{Re}_{u,\textit{roll}} \sim \varGamma _{\textit{roll}}^{-0.61}$. We impose different initial flow conditions (roll structures) with controlled perturbations and demonstrate that the initial condition can influence the final turbulent state. We show that the number of horizontally aligned rolls regulates the global transport: a larger number of rolls induces greater vertical momentum and heat transfer. Our study provides the first systematic experimental evidence of multiple flow states in large aspect ratio turbulent Rayleigh–Bénard convection and clarify how these states influence global transport.

We investigate the oscillation of the large-scale circulation (LSC) in turbulent Rayleigh–Bénard convection by combining laboratory experiments and numerical simulations. The experiments are conducted mainly in a laterally confined rectangular cell, and the flow fields are measured by particle image velocimetry in the vertical mid-plane. It is found that the velocity field exhibits not only the commonly observed oscillation frequencies $(f\tau _o=1,2)$ but also higher-order harmonic frequencies $(f\tau _o\gt 2)$, where $\tau _o$ is the turnover time of the LSC. Spectral proper orthogonal decomposition (SPOD) reveals that the coherent structures underlying these frequencies display an approximate azimuthal periodicity when viewed in polar coordinates. These structures can be interpreted as travelling waves, consistent with the advected-oscillation picture proposed by Brown & Ahlers (2009 J. Fluid Mech., vol. 638, pp. 383–400). A data-driven resolvent analysis, used here as an effective input–output model inferred from the experimental data, yields response modes and gain peaks that are consistent with the SPOD results, and support the interpretation that the dominant oscillations are selected through linear amplification about the mean circulation. This perspective is further supported by direct numerical simulation in a rectangular cell and by velocity measurements in a cylindrical cell. These findings provide an additional viewpoint for understanding the oscillation mechanism of the LSC in turbulent Rayleigh–Bénard convection.

We investigate thermal boundary layer (BL) asymmetry in turbulent Rayleigh–Bénard convection (RBC) under both spherical and annular geometries using different BL theories. Unlike planar RBC, the spherical and annular configurations exhibit asymmetric thermal BLs near the inner and outer boundaries due to boundary curvature and non-uniform radial gravity. We generalise three BL frameworks – the Prandtl–Blasius BL model, the steady free-convective model and the fluctuating BL model – and apply them to both geometries. Direct numerical simulations (DNSs), based on the Oberbeck–Boussinesq equations, are performed in three-dimensional spherical RBC and three-dimensional annular RBC for various radius ratios ($\eta$), gravity profiles and also Prandtl numbers ($ \textit{Pr}$), to compare with the predictions of the extended BL models. We find that the BL asymmetries predicted by both the extended steady free-convective BL and the fluctuating BL agree well with DNS results, with the fluctuating BL model providing the best agreement for the mean temperature profiles. A force-balance analysis further shows that this better performance is consistent with the DNS observation that, in the wall-normal direction within the thermal BL, buoyancy is balanced primarily by the pressure-gradient force. This is consistent with the assumption underlying the steady free-convective and fluctuating BL models. Moreover, the fluctuating BL model explicitly accounts for the contribution of turbulent fluctuations to the heat flux, which further improves its agreement with the DNS mean temperature profiles. We derive analytical expressions for the bulk temperature and the thermal BL thickness ratio as functions of the radius ratio and gravity profile across different Prandtl-number regimes. These expressions are obtained by integrating the similarity thermal equation for both the inner and outer BLs using an approximate similarity streamfunction, and by closing the solutions through a heat-flux matching condition. The resulting leading-order expressions obtained from both the steady free-convective and fluctuating BL models are shown to be the same, and they agree well with DNS data. This analytical result provides a robust and practical tool for quantifying BL asymmetry in curved RBC systems.

Zonal winds on Jovian planets play an important role in governing the cloud dynamics, transport of momentum, scalars and weather patterns. Therefore, it is crucial to understand the evolution of zonal flows and their sustainability. Based on studies in two-dimensional $\beta$-plane set-ups, zonal flow is believed to be forced at the intermediate scale via baroclinic instabilities, and the inverse cascade leads to the transfer of energy to large scales. However, whether such a process exists in three-dimensional deep convection systems remains an open and challenging question. To explore a possible answer, we perform large-eddy simulations at Rayleigh and Ekman numbers in the spans $10^7 {-} 10^{12}$ and $10^{-3}{-}10^{-8}$, respectively, corresponding to the inverse of the convective Rossby number, $1/Ro_c$, ranging from $0.1685$ to $168.52$, in a horizontally rotating Rayleigh–Bénard convection set-up. We find the emergence of mean flow at the expense of small-scale turbulence for $1/Ro_c = 1.68522$, $16.852$ and $168.52$. The turbulent kinetic energy budget analysis shows negative turbulent production in zonal flows, implying an energy transfer from the fluctuating velocity fields to the mean flow. We further quantify this energy transfer in spectral space using the kinetic energy spectra and the energy flux, and conclude that the energy deposited at small scales owing to the work done by buoyancy is transferred to large scales via upscale energy transfer, thus corroborating the emanation of a strong mean flow from chaos.

We present three-dimensional direct numerical simulations of turbulent Rayleigh–Bénard convection in a closed rectangular box whose width $L_y$ and length $L_x$ are 0.8 and 2.4 times the height $H$, respectively. The Rayleigh number $\textit{Ra}$ varies from $10^5$ to $10^{10}$, and the Prandtl number is unity. The advantages of the present configuration are: (a) a relatively stable unidirectional large-scale circulation, consisting of two counter-rotating rolls, fills the cell and fixes the thermal plume ejection- and shear-dominated regions, in contrast to those in closed cylindrical cells. (b) The regions of plume ejection are essentially independent of the sidewalls so that their autonomous existence can be studied. This is because there is some space, or ‘fetch’, for the velocity and thermal boundary layers to develop along the length. (c) This geometry allows one to study the influence of locally thin and thick boundary layers (which follow larger or smaller plume activity) on the scaling of convection properties. In regions of larger plume activity (defined by an incessant movement of plumes), the temperature fluctuation as well as the normalised thermal and viscous dissipation rates decay more slowly with $\textit{Ra}$ than in regions of lower activity. Both viscous and thermal boundary layers thin down rapidly with increasing distance from the plume ejection region. The local thicknesses of both boundary layers decline more rapidly with $\textit{Ra}$ in the ejection region than in regions of impact and shear, where they are similar to each other. Despite these details, the global heat transport laws are practically the same as those in other configurations of low to moderate aspect ratios.

We investigate experimentally the flow structure in a fluid layer of thermal conductivity $k$, heated from below with a uniform heat flux $F$, and exposed to a horizontal temperature gradient, $\lambda$, on the top plate. This is a model system for the dynamics in subglacial lakes where such a competition between Rayleigh–Bénard convection (RBC) and horizontal convection (HC) is thought to happen. We evidence a hysteretic transition from a RBC flow structure to a HC flow structure when the non-dimensional control parameter, $\varLambda = k\lambda /F$, is $4\times 10^{-4}$ when $\varLambda$ is decreasing, and $7\times 10^{-4}$ when $\varLambda$ is increasing. These values are lower than the threshold value found in recent two-dimensional direct numerical simulations (Couston et al. 2022 J. Fluid Mech., vol. 947, p. A13), of order $10^{-2}$, highlighting the importance of exploring three-dimensional dynamics at higher realistic Prandtl numbers, and suggesting that HC may be more common in subglacial lakes than previously predicted. For large values of $\varLambda$, we observe that the warmest part of the top plate has approximately the same temperature as the bottom plate, such that a stable temperature gradient settles below the warm side of the top plate. Thermal plumes are no longer visible in this region, and seem to be replaced by internal gravity waves.

We derive the asymptotic solution for the onset of steady, linear, Boussinesq convection in a rapidly rotating system with stress-free, fixed-flux boundary conditions. While the fixed-temperature (FT) case is attainable analytically with relative ease, the fixed-flux (FF) configuration presents greater complexity. However, in the rapidly rotating limit, the leading-order interior solution remains unaffected by the choice of thermal boundary conditions. We exploit this property by employing an asymptotic approach to characterise the differences between the FT and FF systems. Specifically, this involves constructing a composite boundary layer structure comprising an Ekman layer of thickness $ {\textit{Ta}}^{-1/4}$, where $ \textit{Ta}$ is the Taylor number ($ \textit{Ta} \gg 1$ for rapid rotation), and a thermal boundary layer of thickness $ {\textit{Ta}}^{-1/6}$, to accommodate the FF boundary condition. To capture both scales systematically, we introduce the small parameter ${\varepsilon } = {\textit{Ta}}^{-1/12}$, representing the ratio between the two boundary layer thicknesses, and use it to guide the asymptotic expansion. The asymptotic corrections capturing the differences between the two systems are combined with the FT system to construct the corresponding solution for the FF system. We find an asymptotic correction of ${\mathcal{O}} ( {\textit{Ta}}^{-1/2} )$ to the critical Rayleigh number, corresponding wavenumber, vertical velocity and temperature, along with a correction of ${\mathcal{O}} ( {\textit{Ta}}^{-1/6} )$ to the vertical vorticity.

Turbulence is often treated as memoryless. Once the forcing and control parameters are fixed and after any transients have decayed, the system settles into a unique, statistically stable turbulent state. A growing body of work shows that this paradigm does not have to be true. Even under identical forcing and boundary conditions, turbulent flows may sustain multiple long-lived structures, each with its own characteristic transport properties and fluctuations. The paper by Yao et al. (2026 J. Fluid Mech., vol. 1030, R4) demonstrates this phenomenon particularly clearly for centrifugal convection, where the flow self-organises into different numbers of coherent rolls depending on the initial conditions. Beyond reporting the observation of multiple flow states, they provide a theoretical explanation as to why only certain flow states can exist and why the range of possible multiple states shrinks as turbulence intensifies.