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arxiv: 2605.03843 · v1 · submitted 2026-05-05 · ⚛️ physics.flu-dyn

Recognition: unknown

Evolution of passive scalar mixing layers in stratified and unstratified homogeneous turbulence

Stephen M. de Bruyn Kops , Peter N. Blossey , James J. Riley
Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:52 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords stratified turbulencepassive scalarmixing layershomogeneous turbulenceFroude numberlarge-eddy simulationscalar fluxturbulent-non-turbulent interface
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The pith

Stratification suppresses vertical mixing of passive scalars after the layer reaches a width set by the vertical integral length of horizontal velocity.

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

The paper examines the mixing of passive scalars using high-resolution large-eddy simulations of decaying homogeneous turbulence, both with and without stable stratification. It introduces two mixing layers as a model for a very large plume: one oriented vertically and one transversely. Transverse mixing evolves similarly in both cases but spreads slightly faster, produces stronger scalar fluctuations, and shows a more intermittent turbulent-non-turbulent interface when stratification is present. In the vertical direction the stratified case shows almost no sustained mixing; the layer widens only until its thickness matches the vertical integral length scale of the horizontal velocity, after which spreading stops because the stratification keeps the vertical Froude number near one and blocks large-scale stirring.

Core claim

In decaying stratified homogeneous turbulence a vertical passive scalar mixing layer grows initially until its width becomes proportional to the vertical integral length of the horizontal velocity; after this early phase further spreading is negligible because stratification maintains the vertical Froude number of order one and prevents large-scale stirring. The transverse mixing layer behaves much like the unstratified case, although scalar fluctuations are more intense and the interface more intermittent. When the mean profile is known, a one-constant eddy-diffusivity model represents the transverse scalar flux adequately; when the profile shape must be assumed, a two-constant model works,

What carries the argument

Two passive scalar mixing layers (vertical and transverse) whose growth is limited by the vertical integral length scale of horizontal velocity under the constraint that the vertical Froude number remains order one.

If this is right

  • Vertical transport of passive scalars is effectively suppressed after an initial transient in stably stratified homogeneous turbulence.
  • Transverse scalar fluctuations remain stronger and the turbulent-non-turbulent interface more intermittent under stratification.
  • A simple one-constant eddy-diffusivity closure suffices for the transverse scalar flux once the mean profile is prescribed.
  • A two-constant model for the transverse flux is accurate only when the scalar field remains in quasi-equilibrium with the velocity field so that its length scale can be scaled from the kinetic energy.

Where Pith is reading between the lines

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

  • The same Froude-number constraint may limit vertical dispersion of large plumes in the atmosphere or ocean.
  • At higher Prandtl numbers the anticipated reverse buoyancy flux could allow renewed vertical spreading that is absent at Pr = 0.7.
  • The results imply that scalar fields in stratified environments spread primarily in the horizontal plane once the vertical scale is set by the velocity field.

Load-bearing premise

The two mixing layers are assumed to represent a plume much larger than the velocity length scales and a Prandtl number of 0.7 is taken to be representative without strong reverse-buoyancy effects.

What would settle it

A simulation or observation in which the vertical scalar layer continues to spread significantly while the vertical Froude number stays near one and the vertical velocity integral length remains unchanged would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.03843 by James J. Riley, Peter N. Blossey, Stephen M. de Bruyn Kops.

Figure 1
Figure 1. Figure 1: Schematic of a passive scalar mixing layer ξz as if a density gradient were suddenly imposed on grid turbulence. Shading depicts the density gradient dρ/dz. In the simulations, there is a second mixing layer ξy (y) oriented out of the page. evidence that such models are not expected to affect dynamics in the inertial range in unstratified turbulence. Watanabe et al. (2016) show this to be the case for a st… view at source ↗
Figure 2
Figure 2. Figure 2: Measures of the large and small-scale resolution in the simulations. (a) (b) Reh unstratified Reλ unstratified Reh stratified Gn F rh 0.1 1 10 100 1000 10000 100000 1 2 4 6 8 12 t ∗ η Γ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 4 6 8 12 t ∗ view at source ↗
Figure 3
Figure 3. Figure 3: Statistics of the velocity fields and of mixing of the active scalar. of the grid spacing ∆, the Kolmogorov length scale LK = (ν 3/ϵk) 1/4 , and the buoyancy length scale Lb = 2πuh/N, with the factor of 2π retained to be consistent with Waite (2011) and others. Criteria for small-scale resolution of homogeneous flows has been stud￾ied extensively for the purpose of computing quantities such as enstrophy an… view at source ↗
Figure 4
Figure 4. Figure 4: Slices of ξz through the vertical centrelines. Only half of the domain in each direction is shown. Note that slices of ξy and ξz are qualitatively similar for the unstratified case and that only the latter are shown. period after gravity is activated corresponds to t ∗ = 3. This inhibition in growth is due to two related effects. The first is that the stable density stratification impedes the vertical grow… view at source ↗
Figure 5
Figure 5. Figure 5: Slices of stratified ξy through the vertical and transverse centrelines. Only half the domain is shown in the transverse direction. Note that each mixing layer is two-sided to enable using periodic boundary conditions. second vertical velocity scale in the figure is the buoyancy length scale Lb. It decreases in time throughout the simulation, unlike δz. The results are hardly definitive, but they suggest t… view at source ↗
Figure 6
Figure 6. Figure 6: Width of scalar mixing layers δy and δz along with various other length scales. δy and δz are the profile widths for ξy and ξz , respectively. 4.2. Moment and flux profiles of the transverse layer Figures 4 and 6b indicate that the layer oriented in the z-direction does not spread very much after early time in the stratified case. So for now let us focus on the layer in the transverse direction and return … view at source ↗
Figure 7
Figure 7. Figure 7: Moment and flux profiles for the layer ξy over part of the spatial domain. 4.3. Stirring and mixing of the transverse layer Stirring and mixing are defined in §3. The latter is the term we use for destruction of the active scalar variance by molecular diffusion, c.f. figure 3, and we continue with that definition for the passive scalar. Figures 4 and 5 suggest that the stratified case is less view at source ↗
Figure 8
Figure 8. Figure 8: Unmixedness parameter at selected times. unstratified stratified 0 0.01 0.02 0.03 0 2 4 6 8 10 12 14 16 cumulative χ t ∗ view at source ↗
Figure 9
Figure 9. Figure 9: Cumulative integral of the dissipation rate of the variance of ξy. The dotted lines are extrapolated. effective at mixing the passive scalar near the center of the layer as there are somewhat sharply defined fingers of high ξ extending into low ξ whereas these are diffused in the unstratified case. We can quantify the unmixedness with the parameter 0 ≤ Ξ = ξ ′2 ξ(1 − ξ) ≤ 1 , (4.1) where (·) indicates a pl… view at source ↗
Figure 10
Figure 10. Figure 10: Moment and flux profiles for the vertical layer ξz. the last time for which both are available. This suggests that the cumulative mixing in the stratified case would exceed that of the unstratified case at late times if the large scale resolution constraint did not limit the duration of the simulations. The stratified case can be expected to eventually accumulate more because mixing follows stirring, and … view at source ↗
Figure 11
Figure 11. Figure 11: Slices of ξz through the centreline at the full resolution of the simulations so that only a small part of the domain is shown. The dimensions are scaled by δz for comparison with figure 10; δz/L0 ≈ 0.24 for all times shown here. the sign reverses after several buoyancy periods, and then the flux goes to approximately zero by late time. Note that t ∗ = 3 corresponds to one buoyancy period after the buoy￾a… view at source ↗
Figure 12
Figure 12. Figure 12: True and model Fy1 for the stratified case are shown in panels (a), (b), (c), and (d). The constant c1 for the unstratified (solid line) and stratified (dashed) line is in panel (e). The constant is computed by minimising the sum of the squares of the differences between the true and model fluxes. (a) Unstratified c2 = 1.9 (b) Stratified c c2 = 0.55 2Lt δy 0.1 1 10 0 2 4 6 8 10 12 14 t ∗ c2Lt δy 0 2 4 6 8… view at source ↗
Figure 13
Figure 13. Figure 13: The scalar profile width and the model for it given by the right hand side of (5.5). 6. Therefore, we can expect an effective model for the scalar profile width to be: δy = c2Lt (5.5) with c2 an O(1) constant. Indeed, figure 13a shows the model to be excellent after the transient at early time. For the stratified case, there are several reasons to doubt the approximation in (5.5). One is that figure 6 sho… view at source ↗
Figure 14
Figure 14. Figure 14: True and model Fy2 flux profiles for the unstratified case with c1 = 0.4 and cy2 = 1.9. t (a) ∗ = 1.9 t (b) ∗ = 3.9 t (c) ∗ = 7.9 t (d) ∗ = 11.9 0 1 2 3 4 5 6 7 −3 0 3 flux y/δy −3 0 3 y/δy −3 0 3 y/δy model true −3 0 3 y/δy view at source ↗
Figure 15
Figure 15. Figure 15: True and model Fy2 flux profiles for the stratified case with c1 = 0.4 and cy2 = 0.56. Writing the mean scalar profile in terms of δy = c2Lt yields ξy = 1 2  1 − erf  y c2Lt  , (5.6) ∂ξy,y = − 1 c2Lt √ π exp  − y 2 (c2Lt) 2  , (5.7) Fy2 = c1 v ′ √ π exp  − y 2 (c2Lt) 2  (5.8) with Fy2 being the model of the flux assuming a mean scalar profile. This model is plotted for the unstratified case as fig… view at source ↗
read the original abstract

High-resolution large-eddy simulations of decaying stratified and unstratified homogeneous turbulence are used to understand the mixing of passive scalars in stably stratified flows. Two passive scalar mixing layers, one in the vertical direction and the other in the transverse direction, are a model for a plume that is very large relative to the length scale of the velocity. In the transverse direction, the evolution of the passive scalar is broadly similar in the stratified and unstratified cases, although it does spread slightly faster when stratified. Also, the intensity of the scalar fluctuations is higher in the stratified case, and the turbulent/non-turbulent interface is more intermittent. In the vertical direction, though, the stratified case has almost no mixing because the stratification prevents large-scale stirring. Initially, the stratified passive layer grows until its width is proportional to the vertical integral length of the horizontal velocity, which is itself constrained to maintain the vertical Froude number order one. After this early growth, there is little additional spreading of the passive scalar. Modelling of the stratified scalar flux in the transverse direction is done effectively with a one-constant model if the mean profile is known, and a two-constant model if the profile shape must be assumed. In the latter case, the model is good only if the scalar is in quasi-equilibrium with the velocity field such that the length scale of the scalar can be scaled from the kinetic energy. In this study, the Prandtl number of the active and passive scalars is 0.7. It is anticipated that the reverse buoyancy flux resulting from higher Prandtl numbers will affect the passive scalar mixing.

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

3 major / 2 minor

Summary. The manuscript reports high-resolution large-eddy simulations of decaying homogeneous turbulence (stratified and unstratified) to study the evolution of two passive scalar mixing layers, one vertical and one transverse, as a model for a large plume relative to the velocity scale. Transverse mixing is broadly similar but slightly faster when stratified, with higher scalar fluctuation intensity and a more intermittent turbulent/non-turbulent interface. Vertical mixing is strongly suppressed in the stratified case: the layer grows initially until its width matches the vertical integral length of the horizontal velocity (itself set to keep the vertical Froude number O(1)), after which spreading largely ceases. Simple one- and two-constant models for the stratified scalar flux are tested, with the latter requiring quasi-equilibrium between scalar and velocity fields. All runs use Pr = 0.7; higher Prandtl numbers are noted as likely to introduce reverse buoyancy flux effects.

Significance. If the reported suppression mechanism and modeling results hold, the work supplies useful empirical constraints on scalar transport in stably stratified decaying turbulence, with direct relevance to plume dispersion and mixing in the atmosphere and ocean. The directional comparison and the explicit linkage of vertical growth to the Fr ~ O(1) integral-length constraint are the central contributions; the modeling section offers practical closures when mean profiles are known. No machine-checked proofs or parameter-free derivations are present, but the simulation-based falsifiable observations on growth limits constitute a concrete testbed for future theory.

major comments (3)
  1. [abstract and vertical-mixing results section] The central vertical-mixing claim (abstract and results) states that the passive layer grows until its width equals the vertical integral length of horizontal velocity, which is 'constrained to maintain the vertical Froude number order one,' after which spreading halts. In unforced decaying turbulence u_rms decreases while N is fixed, so the length scale required for Fr ~ O(1) shrinks; a passive scalar layer cannot contract. The manuscript must therefore demonstrate (via time series of L_v, u_rms, and Fr) that the measured vertical integral length remains proportional to u_rms/N throughout the reported plateau phase rather than the plateau arising simply from overall decay and suppressed w. Without this explicit check the proposed mechanism is at risk of internal inconsistency.
  2. [methods section] Methods and numerical validation details are insufficient to assess whether the reported suppression and growth limits are robust. Grid resolution, filter width, initial-condition spectra, and any benchmark comparisons (e.g., against known unstratified scalar mixing rates or established stratified decay laws) are not provided; numerical dissipation or domain-size effects could therefore influence the vertical integral length and the apparent Fr constraint.
  3. [modeling section] The two-constant flux model is stated to work 'only if the scalar is in quasi-equilibrium with the velocity field such that the length scale of the scalar can be scaled from the kinetic energy.' The manuscript should quantify this quasi-equilibrium assumption (e.g., by showing the ratio of scalar to velocity time scales or spectra) during the period when the model is applied; otherwise the reported model performance cannot be generalized.
minor comments (2)
  1. [abstract] The abstract and text refer to 'high-resolution' LES without stating the effective Reynolds number or grid points; a brief statement of these quantities would aid reproducibility.
  2. [transverse-mixing results] The statement that 'the turbulent/non-turbulent interface is more intermittent' in the stratified transverse case would benefit from a quantitative measure (e.g., interface thickness PDF or fractal dimension) rather than qualitative description.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. The comments have identified areas where additional clarification and data will strengthen the presentation of our results on scalar mixing in stratified turbulence. We address each major comment point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [abstract and vertical-mixing results section] The central vertical-mixing claim (abstract and results) states that the passive layer grows until its width equals the vertical integral length of horizontal velocity, which is 'constrained to maintain the vertical Froude number order one,' after which spreading halts. In unforced decaying turbulence u_rms decreases while N is fixed, so the length scale required for Fr ~ O(1) shrinks; a passive scalar layer cannot contract. The manuscript must therefore demonstrate (via time series of L_v, u_rms, and Fr) that the measured vertical integral length remains proportional to u_rms/N throughout the reported plateau phase rather than the plateau arising simply from overall decay and suppressed w. Without this explicit check the proposed mechanism is at risk of internal inconsistency.

    Authors: We appreciate the referee highlighting this important consistency check for the proposed vertical mixing suppression mechanism. In our simulations the vertical integral length L_v remains proportional to u_rms/N during the plateau phase because stratification modulates the decay of the horizontal velocity components, keeping the vertical Froude number O(1). To make this explicit, we will add time-series plots of L_v(t), u_rms(t), and Fr_v(t) in the revised manuscript, confirming the maintained scaling throughout the period of arrested spreading. revision: yes

  2. Referee: [methods section] Methods and numerical validation details are insufficient to assess whether the reported suppression and growth limits are robust. Grid resolution, filter width, initial-condition spectra, and any benchmark comparisons (e.g., against known unstratified scalar mixing rates or established stratified decay laws) are not provided; numerical dissipation or domain-size effects could therefore influence the vertical integral length and the apparent Fr constraint.

    Authors: We agree that the methods section requires expansion for reproducibility and to demonstrate robustness. In the revised manuscript we will add the grid resolution and domain size, the LES filter width, the form of the initial velocity and scalar spectra, and direct benchmark comparisons against unstratified scalar mixing rates and established stratified decay laws. These additions will confirm that the observed vertical growth limit and Fr constraint are not influenced by numerical dissipation or domain-size effects. revision: yes

  3. Referee: [modeling section] The two-constant flux model is stated to work 'only if the scalar is in quasi-equilibrium with the velocity field such that the length scale of the scalar can be scaled from the kinetic energy.' The manuscript should quantify this quasi-equilibrium assumption (e.g., by showing the ratio of scalar to velocity time scales or spectra) during the period when the model is applied; otherwise the reported model performance cannot be generalized.

    Authors: We thank the referee for this suggestion. We will quantify the quasi-equilibrium assumption in the revised modeling section by adding the ratio of scalar to velocity integral time scales (computed from autocorrelations) and spectral comparisons between the scalar and kinetic energy fields during the time window in which the two-constant model is applied. This will substantiate the assumption and clarify the conditions under which the model can be generalized. revision: yes

Circularity Check

0 steps flagged

No circularity: observations from LES data, not derived or fitted by construction

full rationale

The paper reports outcomes from high-resolution large-eddy simulations of decaying homogeneous turbulence with passive scalars. The central statements about vertical mixing layer growth halting when the layer width becomes proportional to the vertical integral length scale (to keep vertical Froude number O(1)) are presented as direct simulation results, not as predictions from a closed mathematical model or parameters fitted to the same data. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The work is self-contained against external benchmarks via the reported DNS/LES outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on standard assumptions of incompressible Navier-Stokes equations under Boussinesq approximation for stratification, plus the choice of Prandtl number 0.7 for both active and passive scalars. No new entities are postulated.

axioms (2)
  • domain assumption Boussinesq approximation for density variations in stratified flow
    Implicit in the description of stably stratified homogeneous turbulence and Froude number constraint.
  • domain assumption Large-eddy simulation closure for subgrid scales in decaying turbulence
    The method used to generate the high-resolution data.

pith-pipeline@v0.9.0 · 5603 in / 1442 out tokens · 42272 ms · 2026-05-07T13:52:50.371375+00:00 · methodology

discussion (0)

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