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REVIEW 4 major objections 5 minor 135 references

Turbulent mesoscale convection in the Boussinesq limit and beyond

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The thermal boundary condition, not the mechanical one, sets the pattern scale in turbulent mesoscale convection.

desk verdict A useful review with some genuinely new DNS, but the headline claim that thermal boundary conditions alone set the pattern scale is not yet as load-bearing as the text suggests; the Neumann 'domain-size' scale is partly a finite-box artifact. read the letter →

arxiv 2501.04809 v1 pith:27ZWU6ZA submitted 2025-01-08 physics.flu-dyn

Shadab Alam , Dmitry Krasnov , Ambrish Pandey , John Panickacheril John , Roshan J. Samuel , Philipp P. Vieweg , Jörg Schumacher This is my paper · ORCID
classification physics.flu-dyn PACS 47.27.te44.25.+f
keywords mesoscaleconvectionturbulentsuperstructuressupergranulesthermalboundaryconditionsconstantheatfluxlowPrandtlnumbercompressiblekineticenergydissipation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This review-plus-extension paper consolidates direct numerical simulations of mesoscale convection in a plane layer heated from below and cooled from above. It argues that the thermal boundary condition controls the formation of long-lived large-scale flow structures: constant-temperature plates give turbulent superstructures with a fixed horizontal scale of about five layer heights, whereas constant-heat-flux plates produce supergranules that aggregate until they fill the whole domain. The paper also reports that about 40% of the near-wall area is coherent shear-dominated motion regardless of aspect ratio, Prandtl number, or Rayleigh number, and it extends the analysis to low Prandtl numbers, compressibility, and the statistics of kinetic energy dissipation. The motivation is that mesoscale convection in the atmosphere and Sun sits between small-scale turbulence and global circulation and must be parametrized in any global model.

What carries the argument

The organizing object is the pair of thermal boundary conditions: the Dirichlet condition of fixed plate temperature versus the Neumann condition of fixed heat flux. In the linear regime the Neumann condition has an infinite critical wavelength $\lambda_{\mathrm{crit}}=\infty$ while the Dirichlet condition has a finite one; the paper shows that this distinction persists far into the turbulent regime and produces the two contrasting classes of long-living large-scale flow structures (TSS vs supergranules). Supporting machinery includes very large horizontally extended DNS domains (aspect ratios up to 60), the decomposition of the dissipation field into solenoidal, dilatational, and inhomogeneous components, and the multifractal analysis of dissipation statistics.

What would settle it

Run a direct numerical simulation of constant-heat-flux convection at aspect ratio 120, matching the Rayleigh and Prandtl numbers of the Γ=60 runs, and track the largest connected coherent temperature pattern over time; if it stops growing before the domain width, the supergranule aggregation is a finite-domain effect and the universal picture fails.

Watch

Extended reading notes

Core claim

The central claim is that thermal boundary conditions determine the large-scale structure formation in turbulent mesoscale convection. For fixed-temperature (Dirichlet) plates the flow organizes into turbulent superstructures (TSS) whose characteristic horizontal extent saturates at about $\Lambda_{\mathrm{TSS}}\approx 5H$ for aspect ratios up to at least 16, while for fixed-heat-flux (Neumann) plates the same flow evolves through a slow aggregation into a supergranule whose final size is the domain width, $\Lambda_{\mathrm{SG}}=\Gamma H$. The mechanical boundary conditions (no-slip vs free-slip) are subdominant for this self-organization. The paper further claims that near-wall coherent regions occupy a universal fraction of roughly 40% independent of $\mathrm{Ra}$, $\mathrm{Pr}$, and $\Gamma$; that low-Prandtl-number convection exhibits a Kolmogorov $k^{-5/3}$ kinetic energy spectrum with superstructure scale approaching about $3H$; and that in compressible convection the kinetic energy dissipation rate is multifractal and more intermittent than in the Oberbeck–Boussinesq limit, with solenoidal and dilatational components concentrated around pre-shock regions.

Load-bearing premise

The conclusions assume that the simulated domains are large enough and the runs long enough that the observed patterns (the 5H superstructure scale and the domain-filling supergranule) reflect intrinsic dynamics rather than finite-size or transient artifacts, even though the supergranule's final size is the domain width by construction.

Editorial extensions

If this is right

  • If thermal boundary conditions set the pattern scale, then global climate and stellar models can use the known boundary type to estimate the size of unresolved mesoscale convection structures without resolving them.
  • Because supergranules aggregate to domain size under constant heat flux, flux-driven convection in nature will have a scale set by the domain, not by fluid properties.
  • Low-Prandtl-number convection (relevant to stars and liquid metals) produces Kolmogorov-type inertial ranges and superstructures of about $3H$, implying structure size varies weakly with $\mathrm{Pr}$ and $\mathrm{Ra}$.
  • The universal ~40% near-wall coherent area fraction offers a simple, robust statistic for wall models and for interpreting experimental boundary-layer measurements.
  • Compressible convection shows enhanced small-scale intermittency relative to the Boussinesq limit even at lower Reynolds numbers, so parametrizations based on Boussinesq turbulence statistics will underestimate extreme dissipation events.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If pattern scale is governed by thermal boundary condition, then observations of pattern scales in natural convection (granules, cloud streets) could be inverted to infer effective boundary conditions at the top and bottom of convective layers.
  • Partially conducting plates interpolate between Dirichlet and Neumann; a testable prediction is that as the plate-to-fluid conductivity ratio decreases, the asymptotic superstructure scale should increase from $\sim 5H$ toward the domain size.
  • The universal 40% coherent fraction suggests that near-wall shear organization is governed by a local instability mechanism independent of the large-scale pattern; if true, the mechanism should be reproducible in a much smaller domain with the same local boundary-layer dynamics.
  • Because supergranule growth is stopped by weak rotation, rotation can serve as a control knob to test the inverse-cascade interpretation; measuring the aggregate scale versus the Rossby number would sharpen the theory.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The manuscript is a review-style article, with some new DNS results, on turbulent mesoscale convection in plane layers. It derives the fully compressible equations and the Boussinesq limit, reports low-Prandtl-number simulations at Pr=0.001, compares Dirichlet and Neumann thermal boundary conditions and different mechanical boundary conditions, examines the effect of rectangular domain shape on turbulent superstructures, and analyzes compressible convection through stratification regimes, temperature-dependent material properties, and the statistics and multifractal properties of the kinetic energy dissipation rate. The central claimed result is that thermal boundary conditions control the large-scale structure formation: constant-temperature plates produce turbulent superstructures with a nearly fixed scale of about 5H, while constant-heat-flux plates produce supergranules that grow to the domain size, with mechanical boundary conditions playing a subdominant role.

Significance. If the central claim holds, the paper identifies a physically important control parameter for the scale of organized convection in planetary and stellar settings, with direct consequences for parametrizations of unresolved mesoscale convection. The paper's strengths include a consistent derivation of the governing equations, a clear framing of compressible convection regimes in the (ε,D) plane, the use of genuinely demanding DNS (for example, more than half a trillion grid points at Pr=0.001), and the connection of the Neumann-case self-organization to linear stability arguments and independent spectral-transfer analyses. The authors are also explicit about several limitations. However, the quantitative support for the central dichotomy and for several accessory universal claims is thin; the Neumann supergranule scale is limited by the domain size by construction, and the Dirichlet saturation scale is not demonstrated by a systematic convergence test. These issues affect the strongest form of the paper's main message.

major comments (4)
  1. [Sec. 4.1, Figs. 6-8] The central claim that Neumann boundary conditions produce supergranules growing to the domain size, whereas Dirichlet conditions produce a converged scale near 5H, is underdetermined exactly where the two cases differ. Because the reported final supergranule scale is ΛSG=ΓH, the final state is by construction the largest scale the finite box permits; the λcrit=∞ linear-stability argument (ref. [76]) establishes that the Neumann base state has no intrinsic wavelength, but it does not demonstrate that in a larger domain the aggregation would continue to the new domain scale on accessible times rather than saturating at some intrinsic intermediate scale. The Dirichlet leg also lacks a convergence test: Sec. 4.1 quotes ΛTSS≈5H for Pr∼1 without a table of ΛTSS versus Γ and run time, and Sec. 3 reports ΛTSS≈3H at Pr≤0.02, so the quoted scale is not Pr-independent. Given the paper's own caveats in Sec. 4.1 ('nor is there evidence that the supergranules vanish beyond certain critical Ra or Pr') and Sec. 7 ('should be considered as a starting point only'), the thermal-boundary-condition dichotomy should be presented as an observation valid for the simulated range Γ≤60, Ra≲10^8, and run times of order 10^4 τf, not as a converged universal law.
  2. [Sec. 3, Table 1 and Fig. 5] The claim that the coherent near-wall area fraction Acoh is always approximately 40% and independent of Prandtl number is not supported as a controlled comparison. The analysis plane is set to x3=δT,rms for Pr=0.7 and to x3=δU,rms for Pr=0.001, i.e., at different wall-normal positions, and the coherent/incoherent classification uses the threshold |u_h|≷u_rms(x3); no sensitivity to this threshold or to the coarse-graining box size is reported. With eleven table entries and no uncertainty estimates on Acoh, the apparent universality could be an artifact of the threshold definition rather than a physical invariant, and the statement in Sec. 7 that Acoh is 'independent of Prandtl number' overreaches the data.
  3. [Sec. 3, eq. (39) and Table 1] The scaling laws Nu∼Ra^0.29 and Re∼Ra^0.5 are fitted to only three Rayleigh numbers (Ra=10^5, 10^6, 10^7) at Pr=0.001, a fact the text itself notes in the preceding sentence. With three points, no assessment of curvature or of a possible regime change is possible, and the low values Nu=1.21, 2.48, 4.57 make a pure power-law fit particularly fragile. These expressions should be labeled as preliminary fits to three points, or supplemented by additional Rayleigh numbers, before they are presented as scaling laws in the summary of the regime.
  4. [Sec. 6.3, Tables 2-4 and Fig. 16] The conclusion that fully compressible convection is more intermittent than Oberbeck-Boussinesq convection rests on two FCC runs and two OB runs without uncertainty estimates for the generalized dimensions D(q) obtained from power-law fits in Fig. 15, and the bulk volume for FCC1 differs from that used for FCC2, OB1, and OB2 (Vb=L^2×[0.2,0.6]H versus L^2×[0.2,0.8]H). The differences between the D(q) curves in Fig. 16 are modest and could be within fit error. Since this intermittency claim is one of the four highlighted 'aspects' of the paper, it should either be presented explicitly as a preliminary trend or be supported by a fit-range and bootstrap sensitivity analysis.
minor comments (5)
  1. [Abstract and Sec. 1] The abstract contains two idiomatic errors: 'are partly not anymore accessible' should be 'are no longer accessible', and 'Beside these experiments' should be 'Besides these experiments'.
  2. [Eq. (20)] The derivative notation in Eq. (20) mixes d and ∂: the first term should be written with ∂T~/∂x3 consistently.
  3. [Table 1] The grid entries such as '5122' and '1002' should be typeset as 512^2 and 100^2; the current notation is confusing, especially when Nbox is defined in the same table.
  4. [Reference [29]] Reference [29] is incomplete: it contains the placeholders 'Flow XXX (2024) XXX. doi:XXX' and needs the actual journal volume, article number, and DOI.
  5. [Sec. 4.1] The sentence 'the critical wavelength is λcrit=2√2 and≈2.02 for free-slip [73] and no-slip conditions [74,75], respectively' is grammatically awkward and should list the two values in separate clauses for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the boundary-condition and structure-formation claims are backed by direct numerical simulations and classical linear-stability results, not by construction or fitted predictions.

full rationale

The paper's central claim, that thermal boundary conditions determine whether large-scale patterns form as turbulent superstructures (~5H for Dirichlet plates) or as supergranules that grow to domain size (Neumann plates), is presented as an empirical finding from DNS, with the supergranule case explicitly described as a domain-limited outcome: “a gradual, time-dependent aggregation eventually leads to a SG with a final size of ΛSG = ΓH” and the paper notes the structure is “affected by the specific domain configuration only at late times when ΛSG/Γ → 1.” The equality ΛSG = ΓH is a consequence of the structure filling the box, but the nontrivial content—that aggregation continues to the domain scale rather than saturating at an intrinsic wavelength—is an empirical DNS observation, not an equation derived from the boundary condition. The Dirichlet ~5H scale is likewise reported from prior direct simulations (e.g., refs. [10,11,13]), not fitted to the conclusion. The linear-stability argument λcrit = ∞ for the Neumann case is attributed to the external classical result of Hurle, Jakeman and Pike (ref. [76]), not to a self-citation. The Acoh ≈ 40% near-wall fraction is a threshold-based diagnostic measured from the simulations, not a parameter tuned to match a target. Although the paper relies substantially on the authors' own prior DNS studies, those are independent published simulations with stated parameters and are externally reproducible/falsifiable; self-citation of earlier computational results is not circular unless the cited result is itself the present claim. The acknowledged limitations (e.g., Sec. 7 calling the work “a starting point only,” and Sec. 4.1 noting no evidence that supergranules vanish beyond certain Ra or Pr) affect the generality of the conclusions, not their logical dependence on inputs. No step in the paper reduces a prediction to its own definition or to a fitted parameter, so the circularity score is 0.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard fluid-mechanics axioms (ideal gas, Boussinesq limit, DNS resolution, multifractal locality) and on several hand-chosen parameters: the material-property exponent, the coherent-area threshold, the bulk averaging volume, and the fitted Nu/Re scaling exponents.

free parameters (5)
  • Power-law exponent beta for k(T), mu(T) = 0, 0.165, 0.5, 7
    Chosen by hand in Sec. 5.2, eq. (45), to explore temperature dependence of transport coefficients; changes heat/momentum transfer by up to 50%.
  • Coherent-area threshold = u_rms (local rms horizontal velocity)
    In Sec. 3, near-wall regions are classified as coherent if |u_h| > u_rms; the reported 40% area fraction depends on this threshold.
  • Nu-Ra scaling exponent = approximately 0.29
    Obtained by power-law fit to three DNS points at Pr=0.001 (Ra=1e5,1e6,1e7); no error bar on the exponent.
  • Re-Ra scaling exponent = approximately 0.5
    Same three-point fit in Sec. 3; reported as a power law without uncertainty.
  • Bulk volume Vb for statistics = L^2 x [0.2,0.8]H, except FCC1: [0.2,0.6]H
    In Sec. 6.2, the bulk averaging volume is chosen per run; the different choice for FCC1 affects the mean dissipation values in Table 4.
assumptions (6)
  • domain assumption Ideal gas equation of state p = R*rho*T and temperature-dependent transport coefficients.
    Used for the fully compressible equations in Sec. 2.1; restricts the review to gas-like fluids, not liquids with complex equations of state.
  • domain assumption Boussinesq approximation valid when epsilon -> 0 then D -> 0 (layer height much smaller than temperature scale height).
    Underlies all RBC/DNS results in Secs. 3-4; the paper derives it from the compressible equations but does not prove its accuracy at the simulated parameters.
  • standard math For the adiabatic equilibrium, a polytropic gas law p ~ rho^gamma is assumed.
    Used to derive profiles (16a-c) in Sec. 2.3; standard in stellar convection models.
  • domain assumption Multifractal formalism and the assumption of local isotropy in the bulk for coarse-grained dissipation.
    Sec. 6.3, eq. (57) and text: local isotropy is assumed in averaging dissipation over balls; the K41 scaling symmetry holds only in the infinite-Reynolds limit.
  • domain assumption DNS resolves all relevant scales down to the viscous and diffusive scales without subgrid modeling.
    Stated in Sec. 1; if under-resolved, the dissipation statistics and scaling exponents would be affected.
  • domain assumption The linear stability result lambda_crit = infinity for Neumann boundary conditions extends to the fully turbulent regime.
    In Sec. 4.1, the explanation of supergranule formation as secondary instabilities relies on the classical long-wavelength instability (refs. [76,71,72]).

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Cite this review

Pith. "Pith review of Turbulent mesoscale convection in the Boussinesq limit and beyond." pith.science (2026). https://pith.science/paper/27ZWU6ZA

@misc{pith2026250104809,
  author       = {Pith},
  title        = {Pith review of: Turbulent mesoscale convection in the Boussinesq limit and beyond},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27ZWU6ZA}},
  note         = {Machine review of arXiv:2501.04809}
}
read the original abstract

Mesoscale convection covers an intermediate scale range between small-scale turbulence and the global organization of the convection flow. It is often characterized by an order of the convection patterns despite very high Rayleigh numbers and strong turbulent fluctuations. In this review, we discuss several aspects of mesoscale convection, which have been investigated by three-dimensional direct numerical simulations. The numerical studies are performed in a characteristic configuration of a plane layer that is heated from below and cooled from above or subject to constant heat flux at the top and bottom boundaries. We discuss the role of the thermal and mechanical boundary conditions for structure formation and study the impact of the domain shape as well as the Prandtl number. With respect to the latter, we focus on low values that arise in astrophysical convection and are partly not anymore accessible in laboratory experiments with liquid metals. Beside these experiments in the Boussinesq approximation, we report studies of non-Boussinesq mesoscale convection. This is done by investigating effects of compressibility and temperature dependence of material properties. The kinetic energy dissipation rate turns out to remain a central quantity for the turbulent mixing in compressible convection. Their different components, statistics, relation to the turbulent viscosity, and the multifractal properties are discussed.

Figures

Figures reproduced from arXiv: 2501.04809 by the authors.

Figure 1
Figure 1. Two prominent examples of mesoscale convection in nature. Left: solar granulation around a sunspot at the surface of Sun. Granules are the manifestation [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Structure of the article in a diagram. Each subtopic, either in the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Parameter plane of compressible convection spanned by superadia [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Mesoscale simulations at Pr = 0.001 in a plane convection layer with Γ = L/H = 25 for three different Rayleigh numbers. The top row shows the streamline plots viewed from above. The bottom row displays the corresponding temperature fields inside the thermal boundary la…
Figure 5
Figure 5. Figure 5: Snapshots of coherent shear-dominated and incoherent shear-free regions near the wall for mesoscale convection at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Impact of di [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 6
Figure 6. Figure 6: Here, a gradual, time-dependent aggregation eventu [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Ubiquity of supergranule formation across the numerically accessible [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Transient gradual supergranule aggregation. The supergranule is the eventual result of a transient gradual aggregation process. The duration of this [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Turbulent superstructures in different rectangular cuboids of Γ1 = 25 for Pr = 7 and Ra = 105 . (a) Γ2 = 1, (b) Γ2 = 3, (c) Γ2 = 5, (d) Γ2 = 10, (e) Γ2 = 15. The temperature field in mid-horizontal x1–x2 plane is displayed. View is always from the top. The flow organiz…
Figure 10
Figure 10. Figure 10: Mean fluctuation profiles of temperature (a,b,c,d) and velocity (e,f,g,h) for di [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: Four different regimes of compressible convection. These are the nearly Oberbeck-Boussinesq (OB), the strongly superadiabatic (SAC), the fully compressible (FCC), and the strongly stratified (SSC) convection regime. (a) Vertical profiles of the rescaled mean superadia…
Figure 12
Figure 12. Figure 12: Snapshot contour plots of different components of the turbulent kinetic energy and kinetic dissipation rate fields taken all at half height x3 = H/2 of the layer for convection flows at Ra = 106 and Pr = 0.7. (a) Solenoidal dissipation rate field ϵs in the Oberbeck-Bo…
Figure 13
Figure 13. Figure 13: Probability density functions (PDFs) of di [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Snapshot contour plots of different quantities taken at half height x3 = H/2 of the convection layer height for FCC2 at Ra = 106 and Pr = 0.7. (a) Solenoidal turbulent viscosity field ν t s . (b) Dilatational turbulent viscosity field ν t d . (c) Local turbulent Mach …
Figure 15
Figure 15. Figure 15: Double logarithmic plots of the sum of the moments of the coarse [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Generalized dimensions (main panels) and related singularity spectra (insets) are shown for di [PITH_FULL_IMAGE:figures/full_fig_p022_16.png]

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Works this paper leans on

135 extracted references · 48 canonical work pages

  1. [76]

    D. T. J. Hurle, E. Jakeman, E. R. Pike, On the solution of the B ´enard problem with boundaries of finite conductivity, Proc. R. Soc. Lond. A 296 (1967) 469–475

    1967

  2. [1]

    L. P. Kadano ff, Turbulent heat flow: Structures and scaling, Phys. Today 54 (2001) 34–39. doi:10.1063/1.1404847

    work page doi:10.1063/1.1404847 2001

  3. [2]

    Ahlers, S

    G. Ahlers, S. Grossmann, D. Lohse, Heat transfer and large scale dy- namics in turbulent Rayleigh-B ´enard convection, Rev. Mod. Phys. 81 (2009) 503–537. doi:10.1103/RevModPhys.81.503

    doi:10.1103/revmodphys.81.503 2009

  4. [3]

    Chill `a, J

    F. Chill `a, J. Schumacher, New perspectives in turbulent Rayleigh- B´enard convection, Eur. Phys. J. E 35 (2012) 58. doi: 10.1140/epje/ i2012-12058-1

    doi:10.1140/epje/ 2012

  5. [4]

    Christensen-Dalsgaard, Helioseismology, Rev

    J. Christensen-Dalsgaard, Helioseismology, Rev. Mod. Phys. 74 (2002) 1073–1129. doi:10.1103/RevModPhys.74.1073

    doi:10.1103/revmodphys.74.1073 2002

  6. [5]

    Rincon, M

    F. Rincon, M. Rieutord, The Sun’s supergranulation, Living Rev. Sol. Phys. 15 (2018) 6. doi:10.1007/s41116-018-0013-5

    doi:10.1007/s41116-018-0013-5 2018

  7. [6]

    Schumacher, K

    J. Schumacher, K. R. Sreenivasan, Colloquium: Unusual dynamics of convection in the Sun, Rev. Mod. Phys. 92 (2020) 041001. doi: 10. 1103/RevModPhys.92.041001

    2020

  8. [7]

    Markson, Atmospheric electrical detection of organized convec- tion, Science 188 (1975) 1171–1177

    R. Markson, Atmospheric electrical detection of organized convec- tion, Science 188 (1975) 1171–1177. doi: 10.1126/science.188. 4194.1171

    arXiv 1975

Show all 135 references
  1. [8]

    B. W. Atkinson, J. Wu Zhang, Mesoscale shallow convection in the atmosphere, Rev. Geophys. 34 (1996) 403–431. doi: 10.1029/ 96RG02623

    1996

  2. [9]

    B. E. Mapes, R. A. Houze Jr., Cloud clusters and superclusters over the warm oceanic pool, Mon. Weather Rev. 121 (1993) 1398–1415. doi:10.1175/1520-0493(1993)121<1398:CCASOT>2.0.CO;2

    1993 doi

  3. [10]

    Pandey, J

    A. Pandey, J. D. Scheel, J. Schumacher, Turbulent superstructures in Rayleigh-B´enard convection, Nat. Commun. 9 (2018) 2118. doi: 10. 1038/s41467-018-04478-0

    2018

  4. [11]

    R. J. A. M. Stevens, A. Blass, X. Zhu, R. Verzicco, D. Lohse, Turbulent thermal superstructures in Rayleigh-B´enard convection, Phys. Rev. Flu- ids 3 (2018) 041501(R). doi:10.1103/PhysRevFluids.3.041501

    2018 doi

  5. [12]

    Green, D

    G. Green, D. G. Vlaykov, J. P. Mellado, M. Wilczek, Resolved energy budget of superstructures in Rayleigh-B´enard convection, J. Fluid Mech. 887 (2020) A21. doi:10.1017/jfm.2019.1008

    2020

  6. [13]

    P. P. Vieweg, J. D. Scheel, J. Schumacher, Supergranule aggregation for constant heat flux-driven turbulent convection, Phys. Rev. Research 3 (2021) 013231. doi:10.1103/PhysRevResearch.3.013231

    2021 doi

  7. [14]

    Ishihara, T

    T. Ishihara, T. Gotoh, Y . Kaneda, Study of high-Reynolds number isotropic turbulence by direct numerical simulation, Annu. Rev. Fluid Mech. 41 (2009) 165–180. doi: 10.1146/annurev.fluid.010908. 165203

    2009 doi

  8. [15]

    Schumacher, A

    J. Schumacher, A. Pandey, V . Yakhot, K. R. Sreenivasan, Transition to turbulence scaling in turbulent Rayleigh-B´enard convection, Phys. Rev. E 98 (2018) 033120. doi:10.1103/PhysRevE.98.033120

    2018 doi

  9. [16]

    Valori, J

    V . Valori, J. Schumacher, Connecting boundary layer dynamics with extreme bulk dissipation events in Rayleigh-B ´enard flow, Europhys. Lett. 134 (2021) 34004. doi:10.1209/0295-5075/134/34004

    2021 doi

  10. [17]

    M. S. Miesch, Large-scale dynamics of the convection zone and tachocline, Living Rev. Sol. Phys. 2 (2005) 1. doi: 10.12942/ lrsp-2005-1

    2005

  11. [18]

    Nordlund, R

    A. Nordlund, R. F. Stein, M. Asplund, Solar surface convection, Living Rev. Sol. Phys. 6 (2009) 2. doi:10.12942/lrsp-2009-2

    2009 doi

  12. [19]

    Stevens, Atmospheric moist convection, Annu

    B. Stevens, Atmospheric moist convection, Annu. Rev. Earth Planet. Sci. 33 (2005) 605–643. doi: 10.1146/annurev.earth.33.092203. 122658

    2005 doi

  13. [20]

    Pauluis, J

    O. Pauluis, J. Schumacher, Self-aggregation of clouds in condition- ally unstable moist convection, Proc. Natl. Acad. Sci. USA 108 (2011) 12623–12628. doi:10.1073/pnas.1102339108

    2011 doi

  14. [21]

    M. K. Verma, Physics of Buoyant Flows, World Scientific, Singapore, 2018

    2018

  15. [22]

    Guerrero, P

    G. Guerrero, P. K. Smolarkiewicz, E. M. de Gouveia dal Pino, A. G. Kosovichev, N. N. Mansour, On the role of tachoclines in solar and stel- lar dynamos, Astrophys. J. 819 (2016) 104. doi:10.3847/0004-637X/ 819/2/104

    2016 doi

  16. [23]

    K. R. Sreenivasan, Turbulent mixing: A perspective, Proc. Natl. Acad. Sci. USA 116 (2019) 18175–18183. doi:10.1073/pnas.1800463115

    2019 doi

  17. [24]

    P. Moin, K. Mahesh, Direct Numerical Simulation: A tool in turbulence research, Annu. Rev. Fluid Mech. 30 (1998) 539–578. doi: 10.1146/ annurev.fluid.30.1.539

    1998

  18. [25]

    P. F. Fischer, An overlapping Schwarz method for spectral element solu- tion of the incompressible Navier-Stokes equations, J. Comp. Phys. 133 (1997) 84–101. doi:10.1006/jcph.1997.5651

    1997

  19. [26]

    J. D. Scheel, M. S. Emran, J. Schumacher, Resolving the fine-scale structure in turbulent Rayleigh-B ´enard convection, New J. Phys. 15 (2013) 113063. doi:10.1088/1367-2630/15/11/113063

    2013 doi

  20. [27]

    P. F. Fischer, S. Kerkemeier, M. Min, Y .-H. Lan, M. Phillips, T. Rath- nayake, E. Merzari, A. Tomboulides, A. Karakus, N. Chalmers, T. War- burton, NekRS, a GPU-accelerated spectral element Navier–Stokes solver, Parallel Comput. 114 (2022) 102982. doi: 10.1016/j.parco. 2022.102982

    2022

  21. [28]

    Sachs, T

    S. Sachs, T. Fr ¨ohlich, J. Schumacher, Suppression of free convec- tion e ffects for spherical 1 kg mass prototype, Int. J. Heat Mass Transfer 170 (2021) 121037. doi:10.1016/j.ijheatmasstransfer. 2021.121037

    2021

  22. [29]

    P. P. Vieweg, T. K ¨aufer, C. Cierpka, J. Schumacher, Digital twin of a large-aspect-ratio Rayleigh-B´enard experiment: Role of thermal bound- ary conditions, measurement errors and uncertainties, Flow XXX (2024) XXX. doi:XXX

    2024

  23. [30]

    Krasnov, O

    D. Krasnov, O. Zikanov, T. Boeck, Comparative study of finite dif- ference approaches in simulation of magnetohydrodynamic turbulence at low magnetic Reynolds number, Comput. Fluids 50 (2011) 46–59. doi:10.1016/j.compfluid.2011.06.015

    2011 doi

  24. [31]

    Baranwal, D

    A. Baranwal, D. A. Donzis, R. D. W. Bowersox, Asymptotic behaviour at the wall in compressible turbulent channels, J. Fluid. Mech. 933 (2022) A28. doi:10.1017/jfm.2021.1087

    2022

  25. [32]

    L. D. Landau, E. M. Lifschitz, Course of Theoretical Physics: Fluid 23 Mechanics, Butterworth Heinemann, Oxford, 1987

    1987

  26. [33]

    Sutherland, The viscosity of gases and molecular force, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 36 (1893) 507–531

    W. Sutherland, The viscosity of gases and molecular force, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 36 (1893) 507–531

  27. [34]

    Verzicco, K

    R. Verzicco, K. R. Sreenivasan, A comparison of turbulent ther- mal convection between conditions of constant temperature and con- stant heat flux, J. Fluid Mech. 595 (2008) 203–219. doi: 10.1017/ S0022112007009135

    2008

  28. [35]

    Johnston, C

    H. Johnston, C. R. Doering, Comparison of turbulent thermal con- vection between conditions of constant temperature and constant flux, Phys. Rev. Lett. 102 (2009) 064501. doi:10.1103/PhysRevLett.102. 064501

    2009 doi

  29. [36]

    Moller, T

    S. Moller, T. K ¨aufer, A. Pandey, J. Schumacher, C. Cierpka, Combined particle image velocimetry and thermometry of turbulent superstructures in thermal convection, J. Fluid Mech. 945 (2022) A22. doi: 10.1017/ jfm.2022.538

    2022

  30. [37]

    Weiss, D

    S. Weiss, D. Schanz, A. O. Erdogdu, A. Schr ¨oder, J. Bosbach, Inves- tigation of turbulent superstructures in Rayleigh–B ´enard convection by Lagrangian particle tracking of fluorescent microspheres, Exp. Fluids 64 (2023) 82. doi:10.1007/s00348-023-03624-9

    2023 doi

  31. [38]

    T. L. Perelman, On conjugated problems of heat transfer, Int. J. Heat Mass Transfer 3 (1961) 293–303. doi: 10.1016/0017-9310(61) 90044-8

    1961 doi

  32. [39]

    C. A. Jones, K. A. Mizerski, M. Kessar, Fully developed anelastic convection with no-slip boundaries, J. Fluid Mech. 930 (2022) A13. doi:10.1017/jfm.2021.905

    2022 doi

  33. [40]

    Verhoeven, T

    J. Verhoeven, T. Wieseh ¨ofer, S. Stellmach, Anelastic versus fully com- pressible turbulent convection Rayleigh-B´enard convection, Astrophys. J. 805 (2015) 62. doi:10.1088/0004-637X/805/1/62

    2015 doi

  34. [41]

    Panickacheril John, J

    J. Panickacheril John, J. Schumacher, Strongly superadiabatic and strat- ified limits of compressible convection, Phys. Rev. Fluids 8 (2023) 103505. doi:10.1103/PhysRevFluids.8.103505

    2023 doi

  35. [42]

    Oberbeck, Ueber die W ¨armeleitung der Fl ¨ussigkeiten bei Ber¨ucksichtigung der Str ¨omungen infolge von Temperaturdi fferenzen, Ann

    A. Oberbeck, Ueber die W ¨armeleitung der Fl ¨ussigkeiten bei Ber¨ucksichtigung der Str ¨omungen infolge von Temperaturdi fferenzen, Ann. Phys. Chem. VII (1879) 271–292

  36. [43]

    Boussinesq, Th ´eorie analytique de la chaleur: mise en harmonie avec la thermodynamique et avec la th ´eorie m ´ecanique de la lumi `ere, Gauthier-Villars, Paris, 1903

    J. Boussinesq, Th ´eorie analytique de la chaleur: mise en harmonie avec la thermodynamique et avec la th ´eorie m ´ecanique de la lumi `ere, Gauthier-Villars, Paris, 1903

    1903

  37. [44]

    Otero, R

    J. Otero, R. W. Wittenberg, R. A. Worthing, C. R. Doering, Bounds on Rayleigh-B´enard convection with an imposed heat flux, J. Fluid Mech. 473 (2002) 191–199. doi:10.1017/S0022112002002410

    2002 doi

  38. [45]

    Pandey, D

    A. Pandey, D. Krasnov, K. R. Sreenivasan, J. Schumacher, Convective mesoscale turbulence at very low Prandtl numbers, J. Fluid Mech. 948 (2022) A23. doi:10.1017/jfm.2022.694

    2022 doi

  39. [46]

    Horanyi, L

    S. Horanyi, L. Krebs, U. M ¨uller, Turbulent Rayleigh–B ´enard convec- tion in low Prandtl–number fluids, Int. J. Heat Mass Transfer 42 (1999) 3983–4003. doi:10.1016/S0017-9310(99)00059-9

    1999 doi

  40. [47]

    Glazier, T

    J. Glazier, T. Segawa, A. Naert, M. Sano, Evidence against ‘ultrahard’ thermal turbulence at very high Rayleigh numbers, Nature 398 (1999) 307–310

    1999

  41. [48]

    V ogt, S

    T. V ogt, S. Horn, A. M. Grannan, J. M. Aurnou, Jump rope vortex in liquid metal convection, Proc. Natl. Acad. Sci. USA 115 (2018) 12674– 12679. doi:10.1073/pnas.1812260115

    2018 doi

  42. [49]

    Z ¨urner, F

    T. Z ¨urner, F. Schindler, T. V ogt, S. Eckert, J. Schumacher, Combined measurement of velocity and temperature in liquid metal convection, J. Fluid Mech. 876 (2019) 1108–1128. doi:10.1017/jfm.2019.556

    2019 doi

  43. [50]

    R. J. Samuel, M. Bode, J. D. Scheel, K. R. Sreenivasan, J. Schumacher, No sustained mean velocity in the boundary region of plane thermal con- vection, J. Fluid Mech. 996 (2024) A49. doi:10.1017/jfm.2024.853

    2024 doi

  44. [51]

    Bailon-Cuba, M

    J. Bailon-Cuba, M. S. Emran, J. Schumacher, Aspect ratio dependence of heat transfer and large-scale flow in turbulent convection, J. Fluid Mech. 655 (2010) 152–173. doi:10.1017/S0022112010000820

    2010 doi

  45. [52]

    J. D. Scheel, J. Schumacher, Predicting transition ranges to fully tur- bulent viscous boundary layers in low Prandtl number convection flows, Phys. Rev. Fluids 2 (2017) 123501. doi:10.1103/PhysRevFluids.2. 123501

    2017 doi

  46. [53]

    R. J. Stevens, R. Hartmann, R. Verzicco, D. Lohse, How wide must Rayleigh–B´enard cells be to prevent finite aspect ratio e ffects in turbu- lent flow?, J. Fluid Mech. 1000 (2024) A58. doi: 10.1017/jfm.2024. 996

    2024 doi

  47. [54]

    Schumacher, P

    J. Schumacher, P. G ¨otzfried, J. D. Scheel, Enhanced enstrophy gen- eration for turbulent convection in low-Prandtl-number fluids, Proc. Natl. Acad. Sci. USA 112 (2015) 9530–9535. doi: 10.1073/pnas. 1505111112

    2015 doi

  48. [55]

    P. P. Vieweg, Large-Scale Flow Structures in Turbulent Rayleigh- B´enard Convection: Dynamical Origin, Formation, and Role in Material Transport, Ph.D. thesis, TU Ilmenau, 2023-12-05. doi:10.22032/DBT. 58334

    2023 doi

  49. [56]

    D. Krug, D. Lohse, R. J. A. M. Stevens, Coherence of temperature and velocity superstructures in turbulent Rayleigh–B´enard flow, J. Fluid Mech. 887 (2020) A2. doi:10.1017/jfm.2019.1054

    2020

  50. [57]

    Foroozani, D

    N. Foroozani, D. Krasnov, J. Schumacher, Turbulent convection for different thermal boundary conditions at the plates, J. Fluid Mech. 907 (2021) A27. doi:10.1017/jfm.2020.830

    2021 doi

  51. [58]

    Zwirner, A

    L. Zwirner, A. Tilgner, O. Shishkina, Elliptical instability and multiple- roll flow modes of the large-scale circulation in confined turbulent Rayleigh-B´enard convection, Phys. Rev. Lett. 125 (2020) 054502. doi:10.1103/Phys.Rev.Lett.125.054502

    2020 doi

  52. [59]

    K. Iyer, J. D. Scheel, J. Schumacher, K. R. Sreenivasan, Classical 1 /3 scaling of convection holds up toRa = 1015, Proc. Natl. Acad. Sci. USA 117 (2020) 7594–7598. doi:10.1073/pnas.1922794117

    2020 doi

  53. [60]

    M. C. Cross, H. S. Greenside, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, Cambridge, UK, 2009

    2009

  54. [61]

    E. L. Koschmieder, B ´enard Cells and Taylor V ortices, Cambridge Uni- versity Press, Cambridge, UK, 1993

    1993

  55. [62]

    Manneville, Rayleigh-B ´enard convection: Thirty years of experimen- tal, theoretical, and modeling work, in: I

    P. Manneville, Rayleigh-B ´enard convection: Thirty years of experimen- tal, theoretical, and modeling work, in: I. Mutabazi, J. E. Wesfreid, E. Guyon (Eds.), Dynamics of Spatio-Temporal Cellular Structures, vol- ume 207, Springer, New York, 2006, pp. 41–65

    2006

  56. [63]

    C. Liu, M. Sharma, K. Julien, E. Knobloch, Fixed-flux Rayleigh–B´enard convection in doubly periodic domains: generation of large-scale shear, J. Fluid Mech. 979 (2024) A19. doi:10.1017/jfm.2023.1057

    2024

  57. [64]

    P. P. Vieweg, J. D. Scheel, R. Stepanov, J. Schumacher, Inverse cascades of kinetic energy and thermal variance in three-dimensional horizontally extended turbulent convection, Phys. Rev. Research 4 (2022) 043098. doi:10.1103/PhysRevResearch.4.043098

    2022 doi

  58. [66]

    P. P. Vieweg, A. Kl¨unker, J. Schumacher, K. Padberg-Gehle, Lagrangian studies of coherent sets and heat transport in constant heat flux-driven turbulent Rayleigh–B ´enard convection, Eur. J. Mech. B-Fluids 103 (2024) 69–85. doi:10.1016/j.euromechflu.2023.08.007

    2024 doi

  59. [67]

    P. P. Vieweg, Supergranule aggregation: A Prandtl number-independent feature of constant heat flux-driven convection flows, J. Fluid Mech. 980 (2024) A46. doi:10.1017/jfm.2024.56

    2024 doi

  60. [68]

    Fodor, J

    K. Fodor, J. P. Mellado, M. Wilczek, On the role of large-scale updrafts and downdrafts in deviations from Monin–Obukhov similarity theory in free convection, Bound.-Layer Meteorol. 172 (2019) 371–396. doi: 10. 1007/s10546-019-00454-3

    2019

  61. [69]

    P. P. Vieweg, C. Schneide, K. Padberg-Gehle, J. Schumacher, La- grangian heat transport in turbulent three-dimensional convection, Phys. Rev. Fluids 6 (2021) L041501. doi: 10.1103/PhysRevFluids.6. L041501

    2021 doi

  62. [70]

    Schneide, P

    C. Schneide, P. P. Vieweg, J. Schumacher, K. Padberg-Gehle, Evolution- ary clustering of Lagrangian trajectories in turbulent Rayleigh–B ´enard convection flows, Chaos 32 (2022) 013123. doi:10.1063/5.0076035

    2022 doi

  63. [71]

    C. J. Chapman, M. R. E. Proctor, Nonlinear Rayleigh–B ´enard convec- tion between poorly conducting boundaries, J. Fluid Mech. 101 (1980) 759–782. doi:10.1017/S0022112080001917

    1980 doi

  64. [72]

    C. J. Chapman, S. Childress, M. R. E. Proctor, Long wavelength thermal convection between non-conducting boundaries, Earth Planet. Sc. Lett. 51 (1980) 362–369. doi:10.1016/0012-821X(80)90217-4

    1980 doi

  65. [73]

    O. M. L. Rayleigh, LIX. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side, Philos. Mag. 32 (1916) 529–546

    1916

  66. [74]

    Pellew, R

    A. Pellew, R. V . Southwell, On maintained convective motion in a fluid heated from below, Proc. R. Soc. Lond. A 176 (1940) 312–343. doi:10. 1098/rspa.1940.0092

    1940

  67. [75]

    Chandrasekhar, Hydrodynamic and hydromagnetic instability, Dover 24 Publications, Mineola, 1961

    S. Chandrasekhar, Hydrodynamic and hydromagnetic instability, Dover 24 Publications, Mineola, 1961

    1961

  68. [77]

    Pikovsky, A

    A. Pikovsky, A. Politi, Lyapunov exponents – A tool to explore complex dynamics, Cambridge University Press, Cambridge, UK, 2016

    2016

  69. [78]

    G. M. Vasil, K. Julien, N. A. Featherstone, Rotation suppresses giant-scale solar convection, Proc. Natl. Acad. Sci. USA 118 (2021) e2022518118. doi:10.1073/pnas.2022518118

    2021 doi

  70. [79]

    Alexakis, R

    A. Alexakis, R. Marino, P. D. Mininni, A. van Kan, R. Foldes, F. Feraco, Large-scale self-organization in dry turbulent atmospheres, Science 383 (2024) 1005–1009. doi:10.1126/science.adg8269

    2024 doi

  71. [80]

    Hartlep, A

    T. Hartlep, A. Tilgner, F. H. Busse, Large scale structures in Rayleigh- B´enard convection at high Rayleigh numbers, Phys. Rev. Lett. 91 (2003) 064501. doi:10.1103/PhysRevLett.91.064501

    2003 doi

  72. [81]

    Hartlep, A

    T. Hartlep, A. Tilgner, F. H. Busse, Transition to turbulent convection in a fluid layer heated from below at moderate aspect ratio, J. Fluid Mech. 554 (2005) 309–322. doi:10.1017/S0022112005006671

    2005 doi

  73. [82]

    M. S. Emran, J. Schumacher, Large-scale mean patterns in turbulent convection, J. Fluid Mech. 776 (2015) 96–108. doi: 10.1017/jfm. 2015.316

    2015 doi

  74. [83]

    P. J. Sakievich, Y . T. Peet, R. J. Adrian, Large-scale thermal motions of turbulent Rayleigh–B´enard convection in a wide aspect-ratio cylindrical domain, Int. J. Heat Fluid Flow 61 (2016) 183–196. doi: 10.1016/j. ijheatfluidflow.2016.04.011

    2016 doi

  75. [84]

    Fr ¨ohlich, P

    J. Fr ¨ohlich, P. Laure, R. Peyret, Large departures from Boussinesq ap- proximation in the Rayleigh-B ´enard problem, Phys. Fluids A 4 (1992) 1355–1372. doi:10.1063/1.858413

    1992 doi

  76. [85]

    S. Horn, O. Shishkina, C. Wagner, On non-Oberbeck–Boussinesq effects in three-dimensional Rayleigh–B ´enard convection in glycerol, J. Fluid Mech. 724 (2013) 175–202. doi:10.1017/jfm.2013.151

    2013 doi

  77. [86]

    Pandey, J

    A. Pandey, J. Schumacher, K. R. Sreenivasan, Non-Boussinesq low- Prandtl-number convection with a temperature-dependent thermal diffu- sivity, Astrophys. J. 907 (2021) 56. doi:10.3847/1538-4357/abd1d8

    2021 doi

  78. [87]

    Pandey, J

    A. Pandey, J. Schumacher, K. R. Sreenivasan, Non-Boussinesq convec- tion at low Prandtl numbers relevant to the Sun, Phys. Rev. Fluids 6 (2021) 100503. doi:10.1103/PhysRevFluids.6.100503

    2021 doi

  79. [88]

    Valori, G

    V . Valori, G. E. Elsinga, M. Rohde, J. Westerweel, T. H. J. J. van der Hagen, Particle image velocimetry measurements of a thermally convective supercritical fluid, Exp. Fluids (2019) 143. doi: 10.1007/ s00348-019-2789-z

    2019

  80. [89]

    Panickacheril John, J

    J. Panickacheril John, J. Schumacher, Compressible turbulent convec- tion in highly stratified adiabatic background, J. Fluid Mech. 972 (2023) R4. doi:10.1017/jfm.2023.724

    2023 doi

  81. [90]

    Alboussi `ere, Y

    T. Alboussi `ere, Y . Ricard, Rayleigh-B ´enard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations, J. Fluid Mech. 817 (2017) 264–305. doi:10.1017/jfm.2017.108

    2017 doi

  82. [91]

    Curbelo, L

    J. Curbelo, L. Duarte, T. Alboussi `ere, F. Dubuffet, S. Labrosse, Y . Ri- card, Numerical solutions of compressible convection with an infi- nite Prandtl number: Comparison of the anelastic and anelastic liquid models with the exact equations, J. Fluid Mech. 873 (2019) 646–68...

    2019 doi

  83. [92]

    H. Yik, V . Valori, S. Weiss, Turbulent Rayleigh-B´3nard convection un- der strong non-Oberbeck-Boussinesq conditions, Phys. Rev. Fluids 5 (2020) 103502. doi:10.1103/PhysRevFluids.5.103502

    2020 doi

  84. [93]

    Macek and G

    M. Macek and G. Zinchenko and V . Musilov´a and P. Urban and J. Schu- macher, Assessing non-Oberbeck-Boussinesq e ffects of convection in cryogenic helium, Phys. Rev. Fluids 8 (2023) 094606. doi: 10.1103/ PhysRevFluids.8.094606

    2023

  85. [94]

    Panickacheril John, J

    J. Panickacheril John, J. Schumacher, Compressible turbulent con- vection: The role of temperature- dependent thermal conductivity and dynamic viscosity, Phys. Fluids 36 (2024) 076132. doi: 10.1063/5. 0216623

    2024 doi

  86. [95]

    Shcheritsa, A

    O. Shcheritsa, A. Getling, O. Mazhorova, E ffects of variable thermal diffusivity on the structure of convection, Phys. Lett. A 382 (2018) 639–

    2018

  87. [96]

    A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers, Dokl. Akad. Nauk SSSR 30 (1941) 9–13

    1941

  88. [97]

    G. K. Batchelor, Small-scale variation of convected quantities like temperature in turbulent fluid .1. General discussion and the case of small conductivity, J. Fluid Mech. 5 (1959) 113–133. doi: 10.1017/ S002211205900009X

    1959

  89. [98]

    Corrsin, On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J

    S. Corrsin, On the spectrum of isotropic temperature fluctuations in an isotropic turbulence, J. Appl. Phys. 22 (1951) 469–473. doi: 10.1063/ 1.1699986

    1951

  90. [99]

    A. N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech. 13 (1962) 82–85. doi:10.1017/ S0022112062000518

    1962

  91. [100]

    A. M. Obukhov, Some specific features of atmospheric turbulence, J. Fluid Mech. 13 (1962) 77–81. doi:10.1017/S0022112062000506

    1962 doi

  92. [101]

    Stolovitzky, P

    G. Stolovitzky, P. Kailasnath, K. R. Sreenivasan, Refined similarity hy- potheses for passive scalars mixed by turbulence, J. Fluid Mech. 297 (1995) 275–291. doi:10.1017/S0022112095003090

    1995 doi

  93. [102]

    Falconer, Fractal Geometry - Mathematical Foundations and Appli- cations, Wiley, Chichester, UK, 2003

    K. Falconer, Fractal Geometry - Mathematical Foundations and Appli- cations, Wiley, Chichester, UK, 2003

    2003

  94. [103]

    Grassberger, On the fractal dimension of the henon attractor, Phys

    P. Grassberger, On the fractal dimension of the henon attractor, Phys. Lett. A 97 (1983) 224–226. doi:10.1016/0375-9601(83)90752-1

    1983 doi

  95. [104]

    H. G. E. Hentschel, I. Procaccia, The infinite number of generalized dimensions of fractals and strange attractors, Physica D 8 (1983) 435–

    1983

  96. [105]

    K. R. Sreenivasan, Fractals and multifractals in fluid turbulence, Annu. Rev. Fluid Mech. 23 (1991) 539–604. doi:10.1146/annurev.fl.23. 010191.002543

    1991

  97. [106]

    Mukherjee, S

    S. Mukherjee, S. D. Murugan, R. Mukherjee, S. S. Ray, Turbulent flows are not uniformly multifractal, Phys. Rev. Lett. 132 (2024) 184002. doi:10.1103/PhysRevLett.132.184002

    2024 doi

  98. [107]

    Schumacher, K

    J. Schumacher, K. R. Sreenivasan, Geometric features of the mixing of passive scalars at high Schmidt numbers, Phys. Rev. Lett. 91 (2003) 174501. doi:10.1103/PhysRevLett.91.174501

    2003 doi

  99. [108]

    Schumacher, K

    J. Schumacher, K. R. Sreenivasan, P. K. Yeung, Very fine structures in scalar mixing, J. Fluid Mech. 531 (2005) 113–122. doi: 10.1017/ S0022112005004040

    2005

  100. [109]

    K. R. Sreenivasan, J. Schumacher, What is the turbulence problem, and when may we regard it as solved?, Annu. Rev. Condens. Matter Phys. 16 (2025) 121–143. doi: 10.1146/ annurev-conmatphys-031620-095842

    2025

  101. [110]

    Gotoh, J

    T. Gotoh, J. Yang, Transition of fluctuations from Gaussian state to turbulent state, Phil. Trans. R. Soc. A. 380 (2022) 20210097. doi: 10. 1098/rsta.2021.0097

    2022

  102. [111]

    Gotoh, T

    T. Gotoh, T. Watanabe, I. Saito, Kinematic e ffects on probability den- sity functions of energy dissipation rate and enstrophy in turbulence, Phys. Rev. Lett. 130 (2023) 254001. doi:10.1103/PhysRevLett.130. 254001

    2023 doi

  103. [112]

    Sarkar, G

    S. Sarkar, G. Erlebacher, M. Y . Hussaini, H. O. Kreiss, The analysis and modelling of dilatational terms in compressible turbulence, J. Fluid Mech. 227 (1991) 473–493. doi:10.1017/S0022112091000204

    1991 doi

  104. [113]

    Sarkar, The pressure-dilatation correlation in compressible flows, Phys

    S. Sarkar, The pressure-dilatation correlation in compressible flows, Phys. Fluids 4 (1992) 2674–2682. doi:10.1063/1.858454

    1992 doi

  105. [114]

    P. G. Huang, G. N. Coleman, P. Bradshaw, Compressible turbulent chan- nel flows: DNS results and modelling, J. Fluid Mech. 305 (1995) 185–

    1995

  106. [115]

    Pirozzoli, F

    S. Pirozzoli, F. Grasso, Direct numerical simulations of isotropic compressible turbulence: Influence of compressibility on dynamics and structures, Phys. Fluids 16 (2004) 4386–4407. doi: 10.1063/1. 1804553

    2004 doi

  107. [116]

    Pirozzoli, Numerical methods for high-speed flows, Annu

    S. Pirozzoli, Numerical methods for high-speed flows, Annu. Rev. Fluid Mech. 43 (2011) 163–194. doi: 10.1146/ annurev-fluid-122109-160718

    2011

  108. [117]

    Panickacheril John, D

    J. Panickacheril John, D. A. Donzis, K. R. Sreenivasan, Does dissipative anomaly hold for compressible turbulence?, J. Fluid Mech. 920 (2021) A20. doi:10.1017/jfm.2021.443

    2021 doi

  109. [118]

    D. A. Donzis, J. Panickacheril John, Universality and scaling in homo- geneous compressible turbulence, Phys. Rev. Fluids 5 (2020) 084609. doi:10.1103/PhysRevFluids.5.084609

    2020 doi

  110. [119]

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

    2000

  111. [120]

    Pandey, D

    A. Pandey, D. Krasnov, J. Schumacher, R. Samtaney, K. R. Sreenivasan, Similarities between characteristics of convective turbulence in confined and extended domains, Physica D 442 (2022) 133537. doi:10.1016/j. physd.2022.133537. 25

    2022

  112. [121]

    Frisch, Turbulence: The Legacy of A

    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov, Cambridge University Press, Cambridge, UK, 1994

    1994

  113. [122]

    E. A. Novikov, R. W. Stewart, Intermittency of turbulence and the spec- trum of fluctuations of energy dissipation, Isv. Akad. Nauk USSR, Ser. Geophys. 3 (1964) 408–413

    1964

  114. [123]

    B. B. Mandelbrot, Intermittent turbulence in self-similar cascades: Di- vergence of high moments and dimension of the carrier, J. Fluid Mech. 62 (1974) 331–358. doi:10.1017/S0022112074000711

    1974 doi

  115. [124]

    Frisch, P.-L

    U. Frisch, P.-L. Sulem, M. Nelkin, A simple dynamical model of inter- mittent fully developed turbulence, J. Fluid Mech. 87 (1978) 719–736. doi:10.1017/S0022112078001846

    1978 doi

  116. [125]

    K. R. Sreenivasan, R. A. Antonia, The phenomenology of small-scale turbulence, Annu. Rev. Fluid Mech. 29 (1997) 435–472. doi:10.1146/ annurev.fluid.29.1.435

    1997

  117. [126]

    Meneveau, K

    C. Meneveau, K. R. Sreenivasan, The multifractal spectrum of the dis- sipation field in turbulent flows, Nucl. Phys. B Proc. Suppl. 2 (1987) 49–76. doi:10.1016/0920-5632(87)90008-9

    1987 doi

  118. [127]

    T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, B. I. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33 (1986) 1141–1151. doi:10.1103/PhysRevA.33. 1141

    1986 doi

  119. [128]

    Brandenburg, Stellar mixing length theory with entropy rain, Astro- phys

    A. Brandenburg, Stellar mixing length theory with entropy rain, Astro- phys. J. 832 (2016) 6. doi:10.3847/0004-637X/832/1/6

    2016 doi

  120. [129]

    Pandey, J

    S. Pandey, J. Schumacher, Reservoir computing model of two- dimensional turbulent convection, Phys. Rev. Fluids 5 (2020) 113506. doi:10.1103/PhysRevFluids.5.113506

    2020 doi

  121. [130]

    Pfe ffer, F

    P. Pfe ffer, F. Heyder, J. Schumacher, Hybrid quantum-classical reservoir computing of thermal convection flow, Phys. Rev. Research 4 (2022) 033176. doi:10.1103/PhysRevResearch.4.033176

    2022 doi

  122. [131]

    Pfe ffer, F

    P. Pfe ffer, F. Heyder, J. Schumacher, Reduced-order modeling of two-dimensional turbulent Rayleigh-B ´enard flow by hybrid quantum- classical reservoir computing, Phys. Rev. Research 5 (2023) 043242. doi:10.1103/PhysRevResearch.5.043242

    2023 doi

  123. [132]

    Kobayashi, K

    K. Kobayashi, K. Fujii, N. Yamamoto, Feedback-driven quantum reservoir computing for time-series analysis, PRX Quantum 5 (2024) 040325. doi:10.1103/PRXQuantum.5.040325

    2024 doi

  124. [133]

    Heyder, J

    F. Heyder, J. P. Mellado, J. Schumacher, Generative convective parametrization of a dry atmospheric boundary layer, J. Adv. Model. Earth Syst. 16 (2024) e2023MS004012. doi:10.1029/2023MS004012. 26

    2024 doi

  125. [218]

    doi:10.1017/S0022112095004599

    doi

  126. [444]

    doi:10.1016/0167-2789(83)90235-X

    doi

  127. [645]

    doi:10.1016/j.physleta.2018.01.009

    2018 doi

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