Convection On/Off Regimes

• Convection on/off regimes are distinct dynamical states in fluid systems, defined by transitions from quiescent (conductive) conditions to vigorous, turbulent convection as control parameters change.
• Research uses analytical frameworks and direct numerical simulations to detail regime transitions, employing metrics like the Rayleigh number, Nusselt scaling, and critical thresholds.
• Understanding these regimes aids in optimizing heat and mass transfer in laboratory, geophysical, and astrophysical settings, with implications for planetary dynamics and climate processes.

Convection on/off regimes refer to distinct dynamical states in thermally or compositionally driven fluid systems, often characterized by transitions between quiescent (conductive or weakly convective) and strongly convective behaviors. These regime boundaries are controlled by various physical parameters—chiefly the Rayleigh number, but also including boundary conditions, rotation, stratification, geometry, material contrasts, and multi-component effects. The classification and analysis of on/off regimes are central to understanding turbulence, pattern formation, planetary dynamics, and heat or mass transfer in laboratory, geophysical, and astrophysical flows. Recent studies have revealed a spectrum of regimes and transitions, including clear markers for regime switching, the role of boundary and bulk instabilities, spatial patterning, and the influence of system inhomogeneities.

1. Regimes in Classic and Modified Rayleigh–Bénard Convection

The canonical Rayleigh–Bénard (RB) configuration—fluid heated from below and cooled from above—provides the reference framework for regime classification. In standard RB cells, convection “off” corresponds to the conductive state below the linear instability threshold, while above this threshold, fluid motion and heat transport “turn on,” with further bifurcations marking transitions from steady to unsteady, chaotic, and ultimately fully turbulent states. The principal parameter is the Rayleigh number, $\mathrm{Ra}$, which compares buoyant forcing to diffusive and viscous damping: $$\mathrm{Ra} = \frac{\alpha \Delta g h^3}{\nu \kappa}$$ where $\alpha$ is thermal expansion, $\Delta$ the temperature difference, $g$ gravity, $h$ cell height, $\nu$ the kinematic viscosity, and $\kappa$ the thermal diffusivity. Secondary parameters, such as the Prandtl number ($\mathrm{Pr} = \nu/\kappa$), aspect ratio, and boundary geometry, modulate the regime boundaries.

In configurations with mixed or spatially varying boundary conditions—such as periodic alternation of insulating and conducting top patches—the regime structure is further enriched (Ripesi et al., 2014, Ostilla-Monico et al., 2020):

• Localized Micro-Convection: At low $\mathrm{Ra}$, spatial inhomogeneity seeds near-boundary micro-cells or rolls, driven by local boundary condition variations. An analytical Laplace dual-series framework captures the exponential decay of their influence into the bulk.
• Bulk Convection Onset: As $\mathrm{Ra}$ increases past a renormalized critical value (dependent on the inhomogeneity lengthscale, $L$, and insulating fraction $\xi$), these localized structures merge into global, domain-spanning rolls, signaling an “on” transition for bulk convective motions.
• Fully Developed and Modulated Turbulence: At higher $\mathrm{Ra}$, pattern scale relative to the thermal boundary layer thickness ($\lambda_T = H/(2\mathrm{Nu})$, with $\mathrm{Nu}$ the Nusselt number) controls the degree to which heat transfer and turbulent structure are modulated by the boundary patterning. If $L \gg \lambda_T$, effective heat extraction is reduced and the transition is delayed.

Table: Impact of Mixed Boundary Conditions on Regimes

Pattern Scale Relative to $\lambda_T$	Convective State	Heat Transfer Modifier
$L \ll \lambda_T$	Bulk turbulence (homogeneous)	$\mathrm{Nu}$ close to uniform
$L \gg \lambda_T$	Locally damped, modulated flow	Reduced $\mathrm{Nu}$, delayed onset

2. Turbulent Transitions and Ultimate Regime

At very large $\mathrm{Ra}$, RB systems exhibit a distinct “Ultimate Regime” marked by changes in both mean transport and fluctuation properties (0905.2755, Roche et al., 2012, Bouillaut et al., 2020):

• Scaling Laws: The standard “hard turbulence” regime exhibits $\mathrm{Nu} \sim \mathrm{Ra}^{1/3}$. The ultimate regime, associated with boundary-layer turbulence, is characterized by an enhanced scaling exponent ($\mathrm{Nu} \sim \mathrm{Ra}^{0.39\text{--}0.44}$) or, in the asymptotic limit, $$\mathrm{Nu} \sim \mathrm{Ra}^{1/2}/(\ln \mathrm{Re})^{3/2}$$ (Kraichnan).
• Transition Triggers: The regime transition threshold can depend sensitively on geometry (aspect ratio scaling as, e.g., $\mathrm{Ra}_U \sim \Gamma^{-2.5}$), as well as sidewall properties—thermal insulation, conductance, or inertia can shift or smooth the transition. Prandtl number effects in the range $\mathrm{Pr}\approx1$–7 are relatively weak for the regime location but non-negligible for precise scaling behavior.
• Statistical Indicators: Transition to the ultimate regime is accompanied by abrupt changes in fluctuation statistics—for example, a jump in the second-order structure function exponent of temperature ($\xi_2$), or heat-flux fluctuation ratios ($\sigma/\langle \mathrm{Nu} \rangle$) (Labarre et al., 2023). Above the transition ($\mathrm{Ra}/\mathrm{Pr} \gtrsim 10^9$), these ratios become insensitive to further increases, indicating turbulent, intermittent boundary layers. Below the threshold, the law-of-large-numbers scaling holds and fluctuations are suppressed.

3. Regimes Induced by Geometry, Rotation, and Multi-Physics

System geometry, rotation, and coupling to additional physical processes generate further regime distinctions:

• Rotating Convection: In spherical shells or planetary configurations, rotation introduces the Ekman number ($E=\nu/(2\Omega H^2)$). Onset and regimes are defined by the rotational constraint (Gastine et al., 2016, Agrawal et al., 14 Aug 2024):
  • Rotation-dominated (“on” for geostrophic convection): $\mathrm{Ra}\ll E^{-8/5}$, enforced by a triple VAC/CIA (viscous/Archimedean/Coriolis or Coriolis/Inertia/Archimedean) balance, with $\mathrm{Nu}\sim\mathrm{Ra}^{3/2}E^2$.
  • Transition to non-rotating-like (“off”) convection as $Ra E^{12/7}$ exceeds the threshold.
  • In systems mimicking the Earth's tangent cylinder, baroclinicity at the lateral boundary region can cause an early breakdown of the Taylor–Proudman constraint and trigger geostrophic turbulence at much lower Rayleigh numbers than in classical setups.
• Coupled Fluid–Porous Systems: Regimes are divided into “deep” (cells traverse both fluid and porous regions) and “shallow” (cells restricted to the fluid) (McCurdy et al., 2022). The transition is set by the depth ratio $d$ and captured by an explicit formula: $$d = \left[\frac{1707.8}{27.1}\, \mathrm{Da}\, \epsilon_T^2 \right]^{1/4}$$ where $\mathrm{Da}$ is the Darcy number and $\epsilon_T$ thermal diffusivity ratio. Numerical simulations show the presence of hybrid “shallow-to-deep” transitions.
• Vertical Convection: Direct numerical simulations reveal at least three regimes (Wang et al., 2020):
  • Regime I: $\mathrm{Ra} \lesssim 5\times 10^{10}$, steady, wall-focused flows with classical scaling exponents.
  • Regime II: $$5\times10^{10} \lesssim \mathrm{Ra} \lesssim 10^{13}$$, onset of unsteadiness and efficient plume ejection.
  • Regime III: $\mathrm{Ra} \gtrsim 10^{13}$, banded zonal flows with altered scaling, enhanced mixing.
• Moist and Double-Diffusive Convection: The release of latent heat or the presence of multiple diffusing species introduces new morphological and dynamical regimes—cellular, funnel, and plume regimes in moist convection (Powers et al., 2 May 2025), and multi-stable branches including convectons, anticonvectons, and their bifurcation “snakes” in doubly diffusive systems (Tumelty et al., 2023, Tumelty et al., 18 Nov 2024).

4. Onset Criteria and Predictors for On/Off Switching

The rigorous identification of regime transitions often relies on analytical and numerically validated predictors:

• Critical Rayleigh numbers: For classical RB, onset occurs at a universal threshold; for modified systems, the critical value is renormalized by geometry, inhomogeneity scales, or stratification parameters.
• Scaling collapse: Data collapse techniques, such as plotting $\mathrm{Nu}(\ell/H)^2$ vs $\mathrm{Ra}(\ell/H)^6$ in radiatively driven convection, reveal master transitions between regimes (Bouillaut et al., 2020).
• Phase diagrams: For coupled fluid–porous systems, phase (or regime) diagrams in pertinent parameter spaces demarcate deep, shallow, and hybrid regions (McCurdy et al., 2022).
• Fluctuation statistics: The ratio $\sigma/\langle \mathrm{Nu} \rangle$ as a function of $\mathrm{Ra}/\mathrm{Pr}$ provides a sharp marker for boundary layer instability and the ultimate regime transition (Labarre et al., 2023).

5. Physical Consequences and Applications

Understanding regime transitions and on/off criteria has substantial implications:

• Heat and Mass Transport: The scaling of the Nusselt number with control parameters determines the efficiency of mixing and transfer, with direct impact on natural (mantle convection, planetary cores, atmospheric and oceanic layers) and engineered systems.
• Planetary and Astrophysical Inference: Extrapolation of laboratory regime diagrams to the low Ekman, large $\mathrm{Ra}$ environments of planetary interiors informs models of core convection, dynamo action, and magnetic field organization (Mather et al., 2020, Gastine et al., 2016, Agrawal et al., 14 Aug 2024).
• Ice Formation and Climate: In lakes, the switch from fully mixed to stratified or decaying convective regimes directly controls the timing and conditions for surface ice formation (Olsthoorn, 1 May 2024).
• Multi-stability and Pattern Transition: The coexistence of multiple localized and extended states in doubly diffusive systems underpins hysteresis and sensitivity to perturbation in natural flows (Tumelty et al., 2023, Tumelty et al., 18 Nov 2024).

6. Regime Complexity Beyond Classical On/Off

Recent experimental and numerical findings indicate that the “on/off” model for convection is often an oversimplification. In real, inhomogeneous, or multi-component systems, a range of intermediate and hybrid regimes occur, and transitions are shaped by system-specific details such as boundary-induced baroclinicity, viscosity contrasts, or stratification. The emergence of sluggish-lid, penetrative, or hybrid deep–shallow convection, or the development of multistability in localized patterning, all exemplify the inherent complexity of convective regime landscapes.

7. Summary Table: Commonly Observed Convection On/Off Regimes

System/Configuration	“Off” Regime	Onset/Transition	“On” Regime	Key Criterion(s)/Parameter(s)
Classical RB Cell (homogeneous)	Conductive (no motion)	Critical $\mathrm{Ra}_c$	Roll convection, turbulent flow	$\mathrm{Ra}$, aspect ratio
RB Cell, Mixed BC (patterned top)	Boundary-localized micro-cells	$\mathrm{Ra}_c'(\xi, L/H)$	Bulk spanning rolls/turbulence	Inhomogeneity scale, insulating fraction
Rotating Spherical Shell	No convection	$\mathrm{Ra}_c(E)$	Rotation–dominated to non-rotating–like flow	Transition in $Ra E^{12/7}$
Internally Heated Layer	Parabolic conduction state	Critical $R_L$	Steady rolls $\to$ turbulence	Rayleigh number $R$
Fluid–Porous Coupled Layer	Conduction	Critical $Ra_m$	Shallow $\leftrightarrow$ deep convection	Depth ratio $d$, Darcy number $Da$
Doubly Diffusive (balanced)	Quiescent conduction	Sub/supercritical $Ra$	Convectons, multistable localized states	Buoyancy ratio $N$
Rotating Moist Convection	Weak up-down symmetric flow	Threshold in $\mathcal{R}$	Cellular $\to$ funnel $\to$ plume regimes	Reduced Rayleigh, Ekman numbers
Vertical Convection	Laminar, wall-constrained flows	$\sim5\times10^{10}$	Zonal banded flows	$\mathrm{Ra}$ for $Pr=10$

• (0905.2755) Temperature fluctuations in the Ultimate Regime of Convection
• (Roche et al., 2012) On the triggering of the Ultimate Regime of convection
• (Ripesi et al., 2014) Natural convection with mixed insulating and conducting boundary conditions: low and high Rayleigh numbers regimes
• (Gastine et al., 2016) Scaling regimes in spherical shell rotating convection
• (Prakash et al., 2016) The role of viscosity contrast on plume structure in laboratory modeling of mantle convection
• (Bershadskii, 2019) Multiplicity of chaotic and turbulent regimes in Rayleigh-Bénard convection
• (Ostilla-Monico et al., 2020) Regime crossover in Rayleigh-Benard convection with mixed boundary conditions
• (Bouillaut et al., 2020) Transition to the ultimate regime in a radiatively driven convection experiment
• (Wang et al., 2020) Regime transitions in thermally driven high-Rayleigh number vertical convection
• (McCurdy et al., 2022) Predicting convection configurations in coupled fluid-porous systems
• (Tumelty et al., 2023) Towards Convectons in the Supercritical Regime: Homoclinic Snaking in Natural Doubly Diffusive Convection
• (Labarre et al., 2023) Heat-flux Fluctuations reveals regime transitions in Rayleigh-Bénard convection
• (Olsthoorn, 1 May 2024) Atmospheric cooling of freshwater near the temperature of maximum density
• (Agrawal et al., 14 Aug 2024) Regimes of rotating convection in an experimental model of the Earth's tangent cylinder
• (Tumelty et al., 18 Nov 2024) Convectons in unbalanced natural doubly diffusive convection
• (Powers et al., 2 May 2025) Morphological Regimes of Rotating Moist Convection
• (Okuda et al., 22 Aug 2025) Convective regimes of internally heated steady thermal convection of temperature-dependent viscous fluid