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arxiv: 1907.02451 · v1 · pith:MOEEXBDZnew · submitted 2019-07-04 · ⚛️ physics.flu-dyn

Sharp transitions in rotating turbulent convection: Lagrangian acceleration statistics reveal a second critical Rossby number

Kim M. J. Alards , Rudie P. J. Kunnen , Richard J. A. M. Stevens , Detlef Lohse , Federico Toschi. , Herman J. H. Clercx This is my paper

Pith reviewed 2026-05-25 09:07 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords rotating Rayleigh-Benard convectionLagrangian tracersacceleration statisticsRossby numberflow structuresNusselt numbercritical transitions
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0 comments X

The pith

Lagrangian acceleration statistics reveal a second critical Rossby number for flow structure change in rotating convection

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

This paper investigates the effect of rotation on turbulent thermal convection by tracking the motion of fluid particles. It establishes that the shift from a large-scale circulation roll to aligned swirling plumes takes place at a Rossby number of about 2.25. This transition is detected by a sharp rise in the root-mean-square value and kurtosis of the horizontal accelerations experienced by tracers close to the top boundary. The well-known increase in heat transport occurs at a higher rotation rate corresponding to Rossby number 2.7. A sympathetic reader would care because it demonstrates that the reorganization of the flow pattern and the improvement in heat transfer are controlled by two separate critical points rather than a single one.

Core claim

In rotating Rayleigh-Benard convection, the dominant flow structure changes from a domain-filling large-scale circulation to a collection of rotation-aligned plumes at a critical Rossby number Ro_c2 ≈ 2.25. This is revealed by the sudden increase in the root-mean-square values and the kurtosis of the horizontal acceleration of Lagrangian tracers near the top plate. This structural transition precedes the sharp transition in the Nusselt number, which occurs at Ro_c1 ≈ 2.7 for the same parameter settings.

What carries the argument

Lagrangian acceleration statistics of tracers near the top plate, with focus on the rms and kurtosis of their horizontal components, which capture small-scale flow properties.

If this is right

  • The flow structure transition occurs at Ro ≈ 2.25, earlier than the heat transfer transition at Ro ≈ 2.7.
  • The large-scale circulation is replaced by aligned plumes at the lower critical value.
  • The boundary layer change from Prandtl-Blasius to Ekman type follows the structural reorganization.
  • Two distinct critical Rossby numbers govern the dynamics at these parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The increase in heat transfer may be a consequence of the plume alignment rather than the driver of the transition.
  • Similar Lagrangian diagnostics could uncover multiple thresholds in other parameter regimes of rotating convection.
  • Experiments measuring particle accelerations near boundaries could confirm the separation of the two critical values.

Load-bearing premise

The abrupt change in tracer acceleration statistics at Rossby number 2.25 corresponds to an independent shift in the dominant flow structures, separate from the Nusselt number transition, and is not due to limitations in the simulation resolution or sampling.

What would settle it

A higher-resolution simulation or laboratory experiment showing continuous variation in horizontal acceleration rms and kurtosis through Rossby number 2.25, or an exact coincidence with the Nusselt transition, would disprove the existence of a distinct second critical Rossby number.

Figures

Figures reproduced from arXiv: 1907.02451 by Detlef Lohse, Federico Toschi., Herman J. H. Clercx, Kim M. J. Alards, Richard J. A. M. Stevens, Rudie P. J. Kunnen.

Figure 1
Figure 1. Figure 1: Sketch of the measurement volumes (not to scale). The gray cube in the center represents a measurement [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: a) The Nusselt number Nu as function of Ro, normalized by Nu(∞) for the non-rotating case. Squares show data from [11, 12] for Ra = 2.73 × 108 and P r = 6.26, circles show data for the current simulations at Ra = 1.3 × 109 and P r = 6.7, and triangles show experimental data by [15] (supplementary data) taken at Ra = 2.19 × 109 and P r = 6.26. Closed symbols are for DNS while the open symbols are for experi… view at source ↗
Figure 1
Figure 1. Figure 1: figure 1. For the horizontal component, [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Root-mean-square (rms) values of the horizontal and vertical acceleration components, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Kurtosis and (b) skewness of the horizontal (squares) and vertical (circles) acceleration statistics as a [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The viscous BL thickness δν/H for cylindrical Rayleigh–B´enard convection with Ra = 1.3 × 109 , P r = 6.7 and Γ = 1. The BL thickness is measured from the DNS as the position of the maximum of the horizontal rms velocity. The horizontal dashed line shows the kinetic BL thickness δν/H = 0.032 and the sloping solid black line shows the theoretical prediction based on the linear Ekman BL theory, where δEk = p… view at source ↗
Figure 6
Figure 6. Figure 6: Rms values of the (a) horizontal and (b) vertical acceleration components, both normalized by the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Joint PDFs of the horizontal Lagrangian velocity of passive tracers, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Joint PDFs of the vertical Lagrangian velocity of passive tracers, [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Joint PDFs of the horizontal Lagrangian acceleration of passive tracers, [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
read the original abstract

In RB convection for fluids with Prandtl number $Pr\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer which can be explained by a change in the character of the BL dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl--Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number $Ro_{c}$ occurs typically in the range $2.3\lesssim Ro_{c}\lesssim 2.9$ (for Rayleigh number $Ra=1.3\times 10^9$, $Pr=6.7$, and a convection cell with aspect ratio $\Gamma=\frac{D}{H}=1$, with $D$ the diameter and $H$ the height of the cell). The explanation of the sharp transition in the heat transfer points to the change in the dominant flow structure. At $1/Ro\lesssim 1/Ro_c$ (slow rotation), the well-known LSC is found: a single domain-filling convection roll made up of many individual thermal plumes. At $1/Ro\gtrsim 1/Ro_c$ (rapid rotation), the LSC vanishes and is replaced with a collection of swirling plumes that align with the rotation axis. In this paper, by numerically studying Lagrangian acceleration statistics, related to the small-scale properties of the flow structures, we reveal that this transition between these different dominant flow structures happens at a second critical Rossby number, $Ro_{c_2}\approx 2.25$ (different from $Ro_{c_1}\approx 2.7$ for the sharp transition in the Nusselt number $Nu$; both values for the parameter settings of our present numerical study). When statistical data of Lagrangian tracers near the top plate are collected, it is found that the root-mean-square (rms) values and the kurtosis of the horizontal acceleration of these tracers show a sudden increase at $Ro_{c_2}$.

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 manuscript reports direct numerical simulations of rotating Rayleigh-Bénard convection (Ra=1.3×10^9, Pr=6.7, Γ=1) and uses Lagrangian tracer statistics to argue for a second critical Rossby number Ro_c2≈2.25. At this value the rms and kurtosis of horizontal acceleration for tracers near the top plate increase abruptly, which the authors interpret as the transition from a domain-filling large-scale circulation to rotation-aligned plumes; this is stated to be distinct from the known Nusselt-number transition at Ro_c1≈2.7.

Significance. If the reported discontinuity is shown to be robust and independent of the heat-transport transition, the result would refine the picture of how rotation reorganizes the flow before the boundary-layer regime changes. The Lagrangian-acceleration diagnostic is a potentially useful probe of small-scale structure that is not directly accessible from Eulerian fields.

major comments (2)
  1. [Abstract / Numerical methods] Abstract and numerical-setup section: no grid resolution, tracer count, integration length, spatial definition of “near the top plate,” or convergence tests are supplied. Because the central claim rests on the sudden jump in rms and kurtosis being physical rather than a sampling or binning artifact, these quantities must be documented and shown to be insensitive to reasonable variations in tracer ensemble size and location.
  2. [Results (tracer statistics)] Results section describing the acceleration statistics: the manuscript must demonstrate that the location of the jump at Ro≈2.25 is statistically distinguishable from Ro_c1≈2.7 and that the same simulations reproduce the known Nu(Ro) transition at the higher value. Without error bars on the acceleration moments or a direct overlay of Nu(Ro) and acceleration(Ro) from identical runs, the claim of two distinct critical values remains under-supported.
minor comments (2)
  1. [Abstract] Abstract: the notation “Prandtl--Blasius” uses an en-dash; standard usage is “Prandtl–Blasius.”
  2. [Abstract] Abstract: the parenthetical remark “both values for the parameter settings of our present numerical study” should be expanded to state explicitly that Ro_c1 was also measured in the same runs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We will revise the paper to address the concerns regarding numerical details and the presentation of the two transitions. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [Abstract / Numerical methods] Abstract and numerical-setup section: no grid resolution, tracer count, integration length, spatial definition of “near the top plate,” or convergence tests are supplied. Because the central claim rests on the sudden jump in rms and kurtosis being physical rather than a sampling or binning artifact, these quantities must be documented and shown to be insensitive to reasonable variations in tracer ensemble size and location.

    Authors: We agree that the numerical parameters and convergence information are necessary to substantiate the claim. The revised manuscript will include these details in the methods section: grid resolution, tracer count, integration length, the spatial definition of the near-plate region, and convergence tests with respect to tracer number and binning. These additions will demonstrate that the jump at Ro_c2 is robust and not due to sampling artifacts. revision: yes

  2. Referee: [Results (tracer statistics)] Results section describing the acceleration statistics: the manuscript must demonstrate that the location of the jump at Ro≈2.25 is statistically distinguishable from Ro_c1≈2.7 and that the same simulations reproduce the known Nu(Ro) transition at the higher value. Without error bars on the acceleration moments or a direct overlay of Nu(Ro) and acceleration(Ro) from identical runs, the claim of two distinct critical values remains under-supported.

    Authors: The manuscript presents data from the same simulations for both the Nusselt number and the Lagrangian statistics. However, to better demonstrate the distinction between the two critical values, we will add error bars to the acceleration statistics and a direct comparison plot of Nu(Ro) and the acceleration moments in the revised version. This will show that the transitions occur at different Rossby numbers. revision: yes

Circularity Check

0 steps flagged

No circularity: critical values extracted directly from simulation data

full rationale

The paper reports an empirical observation from direct numerical simulations: a sudden increase in rms and kurtosis of horizontal Lagrangian acceleration at Ro_c2≈2.25, identified as a distinct transition point separate from the Nu transition at Ro_c1≈2.7. This is presented as a numerical finding from tracer statistics near the top plate, with no equations, fitted parameters, or self-citations that define the reported critical value in terms of itself or reduce the claim to a renaming or construction from prior inputs. The derivation chain consists of simulation output analysis rather than any self-referential structure.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The study rests on standard incompressible Navier-Stokes equations under the Boussinesq approximation for RB convection; the critical Rossby numbers are measured from simulation data rather than introduced as free parameters. No new entities are postulated.

free parameters (3)
  • Rayleigh number Ra = 1.3e9
    Fixed simulation parameter chosen to match prior studies of the heat-transfer transition.
  • Prandtl number Pr = 6.7
    Fixed fluid property chosen for the numerical campaign.
  • Aspect ratio Gamma = 1
    Fixed geometry of the cylindrical cell.
axioms (2)
  • standard math The flow obeys the incompressible Navier-Stokes equations under the Boussinesq approximation.
    Invoked implicitly as the governing equations for all RB convection simulations described in the abstract.
  • domain assumption Lagrangian tracers faithfully sample the acceleration field of the underlying Eulerian flow near the top plate.
    Required for interpreting rms and kurtosis changes as indicators of flow-structure transitions.

pith-pipeline@v0.9.0 · 5981 in / 1596 out tokens · 44188 ms · 2026-05-25T09:07:35.387635+00:00 · methodology

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Reference graph

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