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arxiv: 2605.22326 · v1 · pith:ZOP6QGKEnew · submitted 2026-05-21 · ⚛️ physics.flu-dyn

Lagrangian single-particle, multi-particle and topological analyses in turbulent Rayleigh-B\'enard convection

Matti Ettel , Roshan J. Samuel , Michael Chertkov , J\"org Schumacher This is my paper

Pith reviewed 2026-05-22 02:43 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords Lagrangian turbulenceRayleigh-Bénard convectionparticle dispersionheat transfer intermittencyvelocity gradient topologyplume dynamics
0
0 comments X

The pith

Lagrangian particle tracking reveals extreme intermittency and organized dispersion in Rayleigh-Bénard convection

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper tracks individual fluid particles through direct numerical simulations of Rayleigh-Bénard convection across Rayleigh numbers from 10^5 to 10^10 at Prandtl number 0.7. Heat transport along single trajectories proves far more variable than global averages indicate, with some particles carrying fluxes hundreds of times the mean value. Velocity-gradient invariants along paths mark specific regions tied to vortex stretching and plume release. Pair separation does not settle into one power law; instead a brief rapid ejection phase gives way to sustained scaling, sequenced by buoyancy then shear. These trajectory-based diagnostics therefore isolate the actual transport mechanisms that Eulerian statistics average away.

Core claim

The central claim is that Lagrangian heat transfer remains extremely intermittent, with individual massless particles achieving convective fluxes up to 500 times the global Eulerian mean while higher-order moments relax toward Gaussian values at higher Ra; Q-R analysis of the velocity gradient along trajectories isolates a distinct topological footprint of dust-devil-like convective vortices in the Q>0, R<0 quadrant linked to stretching, detachment, and localized heat transfer; and conditioned pair-dispersion plus cloud PCA show temporally organized transport consisting of a short t^5-like ejection episode followed by sustained Richardson-like t^3 scaling rather than extended plateaus.

What carries the argument

Q-R invariant topology of the velocity gradient tensor sampled along material trajectories together with principal-component analysis of dense Lagrangian particle clouds

Load-bearing premise

The chosen aspect ratio of 4:4:1 and the numerical resolution at Rayleigh numbers up to 10^10 are sufficient to capture the full range of Lagrangian statistics without dominant boundary or discretization artifacts affecting the reported intermittency and dispersion regimes.

What would settle it

A higher-resolution simulation or laboratory experiment at the same or larger Rayleigh numbers that finds substantially lower peak Lagrangian heat fluxes or that eliminates the distinct Q>0, R<0 population would falsify the claimed mechanism-resolving power of the topological and cloud-geometry diagnostics.

Figures

Figures reproduced from arXiv: 2605.22326 by J\"org Schumacher, Matti Ettel, Michael Chertkov, Roshan J. Samuel.

Figure 1
Figure 1. Figure 1: Lagrangian particle acceleration statistics. Probability density functions (PDFs) of the vertical ( [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Evaluation of the Heisenberg-Yaglom constant [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Lagrangian Nusselt number statistics. Probability density functions (PDFs) of the total Lagrangian [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Parameter plane spanned by the invariants [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Analysis of dust-devil-like convective vortices. (a) Projected trajectories ( [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of particle pair and single-particle dispersion. Relative particle pair dispersion [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scale-dependent Lagrangian eddy viscosity [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Geometric overview of particle clouds at [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Geometric evolution of a representative particle cloud seeded in the mixing zone ( [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Ensemble statistics of the temporal evolution of Lagrangian particle clouds. Ensemble averages [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Cubic smoothing splines for Lagrangian particle accelerations at [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
read the original abstract

We present three-dimensional direct numerical simulations of turbulent Rayleigh-B\'enard convection (RBC) in the Lagrangian frame of reference for Rayleigh numbers $10^5 \leq Ra \leq 10^{10}$ and a Prandtl number $Pr=0.7$ in a plane layer at an aspect ratio $L:L:H=4:4:1$ with a horizontal length $L$ and height $H$. We use particle accelerations, Lagrangian heat transfer, $Q$-$R$ invariant topology, Lagrangian particle pair dispersion, scale-dependent Lagrangian eddy viscosity, and principal-component analysis (PCA) of dense particle clouds to characterise convective transport along material trajectories. By computing particle accelerations at the integration time step and controlling spectral-element-method signatures, we obtain robust acceleration statistics and recover Heisenberg-Yaglom behaviour. Lagrangian heat transfer is extremely intermittent: individual massless Lagrangian particles can carry convective heat fluxes up to $500$ times the global Eulerian mean, although higher-order heat flux moments decrease toward Gaussian values with increasing $Ra$. The analysis of velocity gradient invariants in the $Q$-$R$ plane along trajectories identifies a distinct topological footprint of dust-devil-like convective vortices in the quadrant of $Q>0$, $R<0$, associated with vortex stretching, plume detachment, and intense localised heat transfer. Global unconditioned pair dispersion exhibits neither extended Richardson nor Bolgiano-Obukhov scaling plateaus. Rather, scale-dependent eddy viscosity and conditioned PCA of dense particle clouds reveal that buoyancy- and shear-driven dispersion are temporally organised: rapid plume-driven ejection produces a short $t^5$-like episode, followed by sustained Richardson-like $t^3$-scaling. Thus, Lagrangian topology and cloud geometry provide mechanism-resolving diagnostics for active-scalar turbulence beyond RBC-specific global scaling laws.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper presents 3D DNS of turbulent Rayleigh-Bénard convection in the Lagrangian frame for 10^5 ≤ Ra ≤ 10^10 at Pr=0.7 in a 4:4:1 aspect-ratio layer. Lagrangian particle tracking is used to compute accelerations, heat fluxes, Q-R velocity-gradient invariants, pair dispersion, scale-dependent eddy viscosity, and PCA of dense particle clouds. Key results include recovery of Heisenberg-Yaglom acceleration scaling, extreme intermittency with individual particles carrying up to 500 times the mean heat flux, a distinct Q>0, R<0 vortex-stretching footprint linked to plume detachment, and temporally sequenced dispersion (short t^5-like ejection followed by sustained t^3 scaling) revealed only after conditioning. The central claim is that Lagrangian topology and cloud geometry supply mechanism-resolving diagnostics for active-scalar turbulence that go beyond RBC global scaling laws.

Significance. If the reported dispersion regimes, intermittency levels, and topological signatures prove robust to domain size and resolution, the work supplies concrete Lagrangian diagnostics that can distinguish buoyancy-driven ejection from shear-driven spreading in convective turbulence. Explicit recovery of Heisenberg-Yaglom scaling and the Q-R footprint associated with dust-devil-like vortices are strengths that could be useful for model validation and for connecting single-particle statistics to multi-particle cloud evolution.

major comments (2)
  1. [Abstract and simulation-setup description] Abstract and simulation-setup description: the central claim that the short t^5-like ejection followed by sustained t^3 dispersion and the extreme heat-flux intermittency are intrinsic mechanism-resolving features requires that these statistics are insensitive to the chosen aspect ratio 4:4:1 and to the grid resolution at Ra=10^10. In a 4:4:1 domain the large-scale circulation spans a distance comparable to the horizontal box size, which can organize plume ejections and particle-cloud geometry; at Ra=10^10 the Kolmogorov scale is small enough that modest under-resolution of the dissipation range can alter acceleration PDFs and pair-separation curves. No grid-convergence or aspect-ratio sensitivity data are supplied for the conditioned PCA or scale-dependent eddy-viscosity results, leaving open the possibility that the reported temporal organization is partly an artifact of domain or grid
  2. [Dispersion and PCA results] Dispersion and PCA results: the manuscript states that global unconditioned pair dispersion shows neither extended Richardson nor Bolgiano-Obukhov plateaus, yet conditioned PCA of dense particle clouds recovers the t^5-then-t^3 sequence. Because the conditioning is performed post-hoc on plume events identified from the same trajectories, it is unclear whether the temporal organization is an independent diagnostic or is partly selected by the conditioning procedure itself; an a-priori, trajectory-independent definition of the conditioning criterion would be needed to establish that the t^5/t^3 sequence is a genuine dynamical feature rather than a consequence of the analysis choice.
minor comments (2)
  1. [Abstract] The phrase 'controlling spectral-element-method signatures' used to justify robust acceleration statistics is not defined quantitatively; a brief description of the filtering or interpolation procedure applied at the integration time step would clarify how the Heisenberg-Yaglom recovery is achieved.
  2. [Figure captions] Figure captions and axis labels for the Q-R plane plots and the conditioned dispersion curves should explicitly state the number of particles, the time window used for conditioning, and the precise definition of the scale-dependent eddy viscosity to allow direct reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below with clarifications and indicate planned revisions. The responses focus on substance and maintain the integrity of the reported results.

read point-by-point responses

  1. Referee: Abstract and simulation-setup description: the central claim that the short t^5-like ejection followed by sustained t^3 dispersion and the extreme heat-flux intermittency are intrinsic mechanism-resolving features requires that these statistics are insensitive to the chosen aspect ratio 4:4:1 and to the grid resolution at Ra=10^10. In a 4:4:1 domain the large-scale circulation spans a distance comparable to the horizontal box size, which can organize plume ejections and particle-cloud geometry; at Ra=10^10 the Kolmogorov scale is small enough that modest under-resolution of the dissipation range can alter acceleration PDFs and pair-separation curves. No grid-convergence or aspect-ratio sensitivity data are supplied for the conditioned PCA or scale-dependent eddy-viscosity results, leaving open the possibility that the reported temporal organization is partly an artifact of domain or grid

    Authors: We agree that explicit grid-convergence tests and aspect-ratio sensitivity studies for the conditioned PCA and scale-dependent eddy-viscosity statistics are not included in the manuscript. The recovery of Heisenberg-Yaglom scaling in the acceleration PDFs serves as an indirect resolution check, and the 4:4:1 aspect ratio follows standard practice in RBC DNS to accommodate the large-scale circulation. We will revise the manuscript to add a dedicated paragraph discussing resolution criteria (based on Kolmogorov and Batchelor scales) and the consistency of key statistics across the simulated Ra range. We will also explicitly note the absence of aspect-ratio variations as a limitation. New simulations for sensitivity tests are computationally demanding and beyond the scope of the current revision but will be flagged for future work. revision: partial

  2. Referee: Dispersion and PCA results: the manuscript states that global unconditioned pair dispersion shows neither extended Richardson nor Bolgiano-Obukhov plateaus, yet conditioned PCA of dense particle clouds recovers the t^5-then-t^3 sequence. Because the conditioning is performed post-hoc on plume events identified from the same trajectories, it is unclear whether the temporal organization is an independent diagnostic or is partly selected by the conditioning procedure itself; an a-priori, trajectory-independent definition of the conditioning criterion would be needed to establish that the t^5/t^3 sequence is a genuine dynamical feature rather than a consequence of the analysis choice.

    Authors: The conditioning is anchored in the Q-R topological signature (Q>0, R<0 quadrant) and associated intense heat-flux events that we identify as characteristic of plume detachment and dust-devil-like vortices. This criterion is physically motivated by the Eulerian-Lagrangian connection developed in the paper rather than an arbitrary post-selection. Nevertheless, we acknowledge the referee's point that greater transparency is needed. In the revision we will provide an explicit, a-priori definition of the conditioning threshold (including the precise Q-R and heat-flux criteria) and will demonstrate that the t^5-like to t^3 transition remains visible when the same criterion is applied to an independent subset of trajectories not used for topology identification. This will strengthen the claim that the temporal staging is a dynamical feature. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results follow from DNS trajectory post-processing

full rationale

The paper's claims rest on direct numerical simulation of RBC at specified Ra and aspect ratio, followed by post-processing to extract particle accelerations, heat fluxes, Q-R invariants, pair dispersion, scale-dependent eddy viscosity, and PCA of clouds. No derivation step reduces a claimed prediction or diagnostic to a fitted parameter or self-citation by construction; the reported t^5-like ejection, t^3 dispersion, vortex topology, and intermittency emerge as observed statistics from the trajectories. The scale-dependent eddy viscosity is introduced as an analysis tool rather than a self-consistent closure that forces the dispersion result. The work is self-contained against external benchmarks of Lagrangian turbulence statistics and does not rely on load-bearing self-citations or ansatzes imported from prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on standard incompressible fluid equations and numerical methods for DNS; no new free parameters, axioms, or postulated entities are introduced beyond the conventional setup of Rayleigh-Bénard convection.

axioms (2)
  • standard math Incompressible Navier-Stokes equations govern the flow at the simulated Prandtl and Rayleigh numbers
    Implicit foundation of all direct numerical simulations of RBC
  • domain assumption Lagrangian particles faithfully represent material trajectories without significant numerical diffusion or interpolation errors
    Required for all reported acceleration and heat-flux statistics

pith-pipeline@v0.9.0 · 5879 in / 1474 out tokens · 44320 ms · 2026-05-22T02:43:10.039641+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Global unconditioned pair dispersion exhibits neither extended Richardson nor Bolgiano-Obukhov scaling plateaus. Rather, scale-dependent eddy viscosity and conditioned PCA of dense particle clouds reveal that buoyancy- and shear-driven dispersion are temporally organised: rapid plume-driven ejection produces a short t^5-like episode, followed by sustained Richardson-like t^3-scaling.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The analysis of velocity gradient invariants in the Q–R plane along trajectories identifies a distinct topological footprint of dust-devil-like convective vortices in the quadrant of Q>0, R<0

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages

  1. [1]

    & Lohse, D

    Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection. Rev. Mod. Phys. 81, 503--537

    work page 2009

  2. [2]

    & Schumacher, J

    Alam, S., Krasnov, D., Pandey, A., Panickacheril John, J., Samuel, R.J., Vieweg, P.P. & Schumacher, J. 2025 Turbulent mesoscale convection in the Boussinesq limit and beyond. Int. J. Heat Fluid Flow 115, 109856

    work page 2025

  3. [3]

    & Schumacher, J

    Badza, A., Froyland, G., Samuel, R.J. & Schumacher, J. 2026 Convective heat transfer patterns in plane-layer Rayleigh-B\'enard convection revealed by transient almost-invariant sets. Phys. Rev. Res. 8, 023040

    work page 2026

  4. [4]

    & Zenger, C

    Banerjee, S., Krahl, R., Durst, F. & Zenger, C. 2007 Presentation of anisotropy properties of turbulence, invariants versus eigenvalue approaches. J. Turbul. 8, N32

    work page 2007

  5. [5]

    & Mommert, M

    Barta, R., Volk, M.-C., Bauer, C., Wagner, C. & Mommert, M. 2025 Temperature and pressure reconstruction in turbulent Rayleigh–B\'enard convection by Lagrangian velocities using PINN. Meas. Sci. Technol. 36, 075304

    work page 2025

  6. [6]

    & Schumacher, J

    Bhattacharya, S., Krasnov, D., Pandey, A., Gotoh, T. & Schumacher, J. 2026 Scale-resolved turbulent Prandtl number for Rayleigh–B\'enard convection at Pr = 10^ -3 . J. Fluid Mech. 1026, A6

    work page 2026

  7. [7]

    & Roatsch, T

    Bickel, V.T., Almeida, M., Read, M., Schriever, A., Tirsch, D., Hauber, E., Gwinner, K., Thomas, N. & Roatsch, T. 2025 Dust devil migration patterns reveal strong near-surface winds across Mars. Sci. Adv. 11, eadw5170

    work page 2025

  8. [8]

    2015 Turbulent pair dispersion as a ballistic cascade phenomenology

    Bourgoin, M. 2015 Turbulent pair dispersion as a ballistic cascade phenomenology. J. Fluid Mech. 772, 678--704

    work page 2015

  9. [9]

    & Sreenivasan, K.R

    Buaria, D. & Sreenivasan, K.R. 2022 Scaling of acceleration statistics in high Reynolds number turbulence. Phys. Rev. Lett. 128, 234502

    work page 2022

  10. [10]

    & Vulpiani, A

    Boffetta, G., Celani, A., Crisanti, A. & Vulpiani, A. 1999 Pair dispersion in synthetic fully developed turbulence. Phys. Rev. E 60, 6734--6741

    work page 1999

  11. [11]

    & Sokolov, I.M

    Boffetta, G. & Sokolov, I.M. 2002 Statistics of two-particle dispersion in two-dimensional turbulence. Phys. Fluids 14, 3224--3232

    work page 2002

  12. [12]

    1959 Turbulent spectra in a stably stratified atmosphere

    Bolgiano, R. 1959 Turbulent spectra in a stably stratified atmosphere. J. Geophys. Res. 64, 2226--2229

    work page 1959

  13. [13]

    1903 Théorie Analytique de La Chaleur

    Boussinesq, J.V. 1903 Théorie Analytique de La Chaleur. vol. 2 (Gauthier-Villars)

    work page 1903

  14. [14]

    & Lebedev, V

    Chertkov, M., Falkovich, G., Kolokolov, I. & Lebedev, V. 1995 Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar. Phys. Rev. E 52, 4924--4941

    work page 1995

  15. [15]

    & Shraiman, B.I

    Chertkov, M., Pumir, A. & Shraiman, B.I. 1999 Lagrangian tetrad dynamics and the phenomenology of turbulence. Phys. Fluids 11, 2394--2410

    work page 1999

  16. [16]

    & Schumacher, J

    Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh-Bénard convection. Eur. Phys. J. E 35, 58

    work page 2012

  17. [17]

    & Cantwell, B.J

    Chong, M.S., Perry, A.E. & Cantwell, B.J. 1990 A general classification of three‐dimensional flow fields. Phys. Fluids 2, 765--777

    work page 1990

  18. [18]

    & Oliva, A

    Dabbagh, F., Trias, F.X., Gorobets, A. & Oliva, A. 2016 On the evolution of flow topology in turbulent Rayleigh-Bénard convection. Phys. Fluids 28, 115105

    work page 2016

  19. [19]

    & Padberg, K

    Dellnitz, M., Froyland, G., Horenkamp, C. & Padberg, K. 2009 On the approximation of transport phenomena a dynamical systems approach. GAMM-Mitt. 32, 47--60

    work page 2009

  20. [20]

    2022 NekRS, a GPU-accelerated spectral element Navier–Stokes solver

    Fischer, P.F. 2022 NekRS, a GPU-accelerated spectral element Navier–Stokes solver. Parallel Comput. 114, 102982

    work page 2022

  21. [21]

    & Pinton, J.-F

    Gasteuil, Y., Shew, W.L., Gibert, M., Chillà, F., Castaing, B. & Pinton, J.-F. 2007 Lagrangian temperature, velocity, and local heat flux measurement in Rayleigh-Bénard convection. Phys. Rev. Lett. 99, 234302

    work page 2007

  22. [22]

    & Kupiainen, A

    Gaw e dzki, K. & Kupiainen, A. 1995 Anomalous scaling of the passive scalar. Phys. Rev. Lett. 75, 3834--3837

    work page 1995

  23. [23]

    & Raasch, S

    Giersch, S. & Raasch, S. 2021 Evolution and features of dust devil-like vortices in turbulent Rayleigh-Bénard convection — A numerical study using direct numerical simulation. J. Geophys. Res.: Atmospheres 126, 7

    work page 2021

  24. [24]

    & Tuzhilina, E

    Greenacre, M., Groenen, P.J.F., Hastie, T., D'Enza, A.I., Markos, A. & Tuzhilina, E. 2022 Principal component analysis. Nat. Rev. Methods Primer 2, 100

    work page 2022

  25. [25]

    2015 Lagrangian Coherent Structures

    Haller, G. 2015 Lagrangian Coherent Structures. Annu. Rev. Fluid Mech. 47, 137--162

    work page 2015

  26. [26]

    1948 Zur statistichen Theorie der Turbulenz

    Heisenberg, W. 1948 Zur statistichen Theorie der Turbulenz. Z. Phys. 124, 628--657

    work page 1948

  27. [27]

    2002 Scaling of acceleration in locally isotropic turbulence

    Hill, R.J. 2002 Scaling of acceleration in locally isotropic turbulence. J. Fluid Mech. 452, 361--370

    work page 2002

  28. [28]

    & Bodenschatz, E

    Jucha, J., Xu, H., Pumir, A. & Bodenschatz, E. 2014 Time-reversal-symmetry breaking in turbulence. Phys. Rev. Lett. 113, 054501

    work page 2014

  29. [29]

    & du Puits, R

    Kästner, C., Schneider, J.D. & du Puits, R. 2023 Evolution and features of dust devil-Like vortices in turbulent Rayleigh-B\'enard convection--An experimental study. J. Geophys. Res.: Atmospheres 128, e2022JD037466

    work page 2023

  30. [30]

    1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers

    Kolmogorov, A.N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301--304

    work page 1941

  31. [31]

    & du Puits, R

    Lösch, A. & du Puits, R. 2021 The Barrel of Ilmenau: A large-scale convection experiment to study dust devil-like flow structures. Meteorol. Z. (Contrib. Atm. Sci.) 30, No.1, 89--97

    work page 2021

  32. [32]

    & Shishkina, O

    Lohse, D. & Shishkina, O. 2024 Ultimate Rayleigh-Bénard turbulence. Rev. Mod. Phys. 96, 035001

    work page 2024

  33. [33]

    & Newman, G.R

    Lumley, J.L. & Newman, G.R. 1977 The return to isotropy of homogeneous turbulence. J. Fluid Mech. 82, 161--178

    work page 1977

  34. [34]

    1978 Computational modeling of turbulent flows

    Lumley, J.L. 1978 Computational modeling of turbulent flows. Adv. Appl. Math. 18, 123--176

    work page 1978

  35. [35]

    & Houze, R.A

    Mapes, B.E. & Houze, R.A. 1993 Cloud clusters and superclusters over the oceanic warm pool. Mon. Weather Rev. 121, 1398--1416

    work page 1993

  36. [36]

    & Xia, K.-Q

    Ni, R., Huang, S.-D. & Xia, K.-Q. 2012 Lagrangian acceleration measurements in convective thermal turbulence. J. Fluid Mech. 692, 395--419

    work page 2012

  37. [37]

    1879 Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen

    Oberbeck, A. 1879 Ueber die Wärmeleitung der Flüssigkeiten bei Berücksichtigung der Strömungen infolge von Temperaturdifferenzen. Ann. Phys. Chem. 243, 271--292

    work page

  38. [38]

    Obukhov, A. M. 1959 On influence of buoyancy forces on the structure of temperature field in a turbulent flow. Dokl. Akad. Nauk SSSR 125, 1246--1248

    work page 1959

  39. [39]

    Soria, J

    Ooi, A., Martin, J. Soria, J. & Chong, M.S. 1999 A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence. J. Fluid Mech. 381, 141--174

    work page 1999

  40. [40]

    & Schumacher, J

    Pandey, A., Scheel, J.D. & Schumacher, J. 2018 Turbulent superstructures in Rayleigh-B\'enard convection. Nat. Commun. 9, 2118

    work page 2018

  41. [41]

    2000 Turbulent Flows

    Pope, S.B. 2000 Turbulent Flows. Cambridge University Press

    work page 2000

  42. [42]

    & Wilkinson, M

    Pumir, A. & Wilkinson, M. 2011 Orientation statistics of small particles in turbulence. New J. Phys. 13, 093030

    work page 2011

  43. [43]

    1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side

    Rayleigh, L. 1916 On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Lond. Edinburgh Dublin Phil. Mag. J. Sci. 32, 529--546

    work page 1916

  44. [44]

    1926 Atmospheric diffusion shown on a distance-neighbour graph

    Richardson, L.F. 1926 Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond. A 110, 709--737

    work page 1926

  45. [45]

    & Schumacher, J

    Samuel, R.J., Bode, M., Scheel, J.D., Sreenivasan, K.R. & Schumacher, J. 2024 No sustained mean velocity in the boundary region of plane thermal convection. J. Fluid Mech. 996, A49

    work page 2024

  46. [46]

    & Schumacher, J

    Samuel, R.J. & Schumacher, J. 2025 Plane-layer Rayleigh–B\'enard convection up to =10^ 11 : near-wall fluctuations and role of initial conditions. Fluid Dyn. Res. 57, 061401

    work page 2025

  47. [47]

    M., Bodenschatz, E

    Sawford, B.L., Yeung, P.K., Borgas, M.S., Vedula, P., La Porta, A., Crawford, A. M., Bodenschatz, E. 2003 Conditional and unconditional acceleration statistics in turbulence. Phys. Fluids 15, 3478–-3489

    work page 2003

  48. [48]

    & Schumacher, J

    Scheel, J.D., Emran, M.S. & Schumacher, J. 2013 Resolving the fine-scale structure in turbulent Rayleigh–Bénard convection. New J. Phys. 15, 113063

    work page 2013

  49. [49]

    2008 Lagrangian dispersion and heat transport in convective turbulence

    Schumacher, J. 2008 Lagrangian dispersion and heat transport in convective turbulence. Phys. Rev. Lett. 100, 134502

    work page 2008

  50. [50]

    2009 Lagrangian studies in convective turbulence

    Schumacher, J. 2009 Lagrangian studies in convective turbulence. Phys. Rev. E 79, 056301

    work page 2009

  51. [51]

    & Sreenivasan, K.R

    Schumacher, J. & Sreenivasan, K.R. 2020 Colloquium: Unusual dynamics of convection in the Sun. Rev. Mod. Phys. 92, 041001

    work page 2020

  52. [52]

    & Schumacher, J

    Schneide, C., Pandey, A., Padberg-Gehle, K. & Schumacher, J. 2018 Probing turbulent superstructures in Rayleigh-Bénard convection by Lagrangian trajectory clusters. Phys. Rev. Fluids 3, 113501

    work page 2018

  53. [53]

    & Padberg-Gehle, K

    Schneide, C., Vieweg, P.P., Schumacher, J. & Padberg-Gehle, K. 2022 Evolutionary clustering of Lagrangian trajectories in turbulent Rayleigh–Bénard convection flows. Chaos 32, 013123

    work page 2022

  54. [54]

    & Schumacher, J

    Shevkar, P.P., Samuel, R.J., Cierpka, C. & Schumacher, J. 2025 Three-dimensional velocity gradient statistics in a mesoscale convection laboratory experiment. J. Fluid Mech. 1024, A47

    work page 2025

  55. [55]

    & Sreenivasan, K.R

    Shevkar, P.P., Samuel, R.J., Zinchenko, G., Bode, M., Schumacher, J. & Sreenivasan, K.R. 2025 Hierarchical network of thermal plumes and their dynamics in turbulent Rayleigh–B\'enard convection. Proc. Natl. Acad. Sci. USA 122, e2502972122

    work page 2025

  56. [56]

    & Siggia, E.D

    Shraiman, B.I. & Siggia, E.D. 1998 Anomalous scaling for a passive scalar near the Batchelor limit. Phys. Rev. E 57, 2965--2977

    work page 1998

  57. [57]

    2018 Turbulent mixing: A perspective

    Sreenivasan, K.R. 2018 Turbulent mixing: A perspective. Proc. Natl. Acad. Sci. USA 116, 18175--18183

    work page 2018

  58. [58]

    & Verzicco, R

    Stevens, R.J.A.M., Lohse, D. & Verzicco, R. 2011 Prandtl and Rayleigh number dependence of heat transport in high Rayleigh number thermal convection. J. Fluid Mech. 688, 31--43

    work page 2011

  59. [59]

    & Lohse, D

    Stevens, R.J.A.M., Blass, A., Zhu, X., Verzicco, R. & Lohse, D. 2018 Turbulent thermal superstructures in Rayleigh-B\'enard convection. Phys. Rev. Fluids 3, 041501(R)

    work page 2018

  60. [60]

    1921 Diffusion by continuous movements

    Taylor, G.I. 1921 Diffusion by continuous movements. Proc. Lond. Math. Soc. 2(20), 196--212

    work page 1921

  61. [61]

    & Verma, M.K

    Tiwari, H.,Sharma, L. & Verma, M.K. 2025 On the absence of the ultimate regime in turbulent thermal convection. Proc. Natl. Acad. Sci. USA 122, e2513474122

    work page 2025

  62. [62]

    & Bodenschatz, E

    Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375--404

    work page 2009

  63. [63]

    2018 Physics of buoyant flows: From instabilities to turbulence

    Verma, M.K. 2018 Physics of buoyant flows: From instabilities to turbulence. World Scientific Publishing Company, Singapore

    work page 2018

  64. [64]

    1982 Local interaction between vorticity and shear in a perfect incompressible fluid

    Vieillefosse, P. 1982 Local interaction between vorticity and shear in a perfect incompressible fluid. J. Phys 43, (6) 837--842

    work page 1982

  65. [65]

    P., Klünker, A., Schumacher, J

    Vieweg, P. P., Klünker, A., Schumacher, J. & Padberg-Gehle, K. 2024 Lagrangian studies of coherent sets and heat transport in constant heat flux-driven turbulent Rayleigh–Bénard convection. Eur. J. Mech. B-Fluids 103, 69--85

    work page 2024

  66. [66]

    & Bosbach, J

    Weiss, S., Schanz, D., Erdogdu, A.O., Schröder, A. & Bosbach, J. 2024 On Lagrangian properties of turbulent Rayleigh–Bénard convection. J. Fluid Mech. 999, A90

    work page 2024

  67. [67]

    & Boffetta, G

    Xu, H., Pumir, A., Falkovich, G., Bodenschatz, E., Shats, M., Xia, H., Francois, N. & Boffetta, G. 2014 Flight–crash events in turbulence. Proc. Natl. Acad. Sci. USA 111(21), 7558--7563

    work page 2014

  68. [68]

    Yaglom, A. M. 1949 On the acceleration field in a turbulent flow. Dokl. Akad. Nauk SSSR 67(5), 795--798

    work page 1949

  69. [69]

    Yeung, P. K. 2002 Lagrangian investigations of turbulence. Annu. Rev. Fluid Mech. 34, 115--142

    work page 2002

  70. [70]

    & Read, P

    Young, R. & Read, P. 2017 Forward and inverse kinetic energy cascades in Jupiter’s turbulent weather layer. Nat. Phys. 13, 1135--1140

    work page 2017