Early onset of secondary shear instability in Kelvin-Helmholtz braids at high Reynolds number

Adrien Lefauve; Emma R. Bouckley; Sam F. Lewin

arxiv: 2604.16173 · v1 · submitted 2026-04-17 · ⚛️ physics.flu-dyn

Early onset of secondary shear instability in Kelvin-Helmholtz braids at high Reynolds number

Emma R. Bouckley , Sam F. Lewin , Adrien Lefauve This is my paper

Pith reviewed 2026-05-10 07:09 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn

keywords Kelvin-Helmholtz instabilitysecondary shear instabilitystratified shear flowbraid regionshigh Reynolds numberdiapycnal mixingbaroclinic shear

0 comments

The pith

Secondary shear instabilities develop early in Kelvin-Helmholtz braids at high Reynolds numbers while primary billows continue growing.

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

The paper develops an inviscid time-dependent model for the braid regions between primary Kelvin-Helmholtz billows by modifying the Corcos-Sherman analysis and adding a stability criterion based on the ratio of strain rate to shear. It shows that this instability criterion can be met significantly earlier than primary billow saturation when the initial Richardson number is high enough, because stronger stratification slows the billow growth but speeds up baroclinic shear production in the braid. Two-dimensional direct numerical simulations at Reynolds numbers up to 10^7 confirm that at high Re the secondary shear instability appears early in the braid, before viscosity can slow the thinning. A sympathetic reader would care because the result offers a mechanistic account for why mixing in stratified shear flows often appears braid-dominated in observations and because it implies that this early instability can set the path to three-dimensional turbulence and diapycnal mixing.

Core claim

At sufficiently high initial Richardson number, the additional stability criterion is satisfied early in the braid while the primary billow is still growing; two-dimensional DNS up to Re=10^7 show that at high Re this early onset indeed occurs before viscous effects slow braid thinning, so that secondary shear instability precedes both vortex pairing and secondary convective instabilities inside the billow core.

What carries the argument

Inviscid time-dependent braid model in braid-aligned coordinates that incorporates the ratio of strain rate to shear as an extra stability criterion alongside the classical Corcos-Sherman analysis.

If this is right

At geophysically relevant Ri and Re, secondary shear instability controls the route to three-dimensional turbulent transition and the resulting diapycnal mixing.
The early braid instability precedes and can pre-empt both vortex pairing instabilities and secondary convective instabilities inside the billow core.
The timing supplies a mechanistic explanation for field observations in which mixing appears dominated by braid activity rather than billow-core activity.

Where Pith is reading between the lines

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

Revised parameterizations of ocean or atmospheric mixing may need to include early braid instabilities to capture the correct timing and efficiency of diapycnal transport.
The model suggests that increasing the initial stratification can advance the onset of secondary instability relative to primary billow growth, a dependence that could be checked in controlled laboratory experiments at intermediate Re.
If the early-onset result survives in three dimensions, braid-focused mixing diagnostics could be used to infer local Reynolds numbers from observed instability timing.

Load-bearing premise

The ratio-of-strain-to-shear stability criterion accurately predicts onset inside the inviscid time-dependent model and this prediction remains valid before viscous or three-dimensional effects become important.

What would settle it

A three-dimensional simulation or field measurement at Re greater than or equal to 10^6 that shows whether secondary shear instability begins before the primary billow saturates or whether viscosity and three-dimensional motions suppress the early onset predicted by the model.

Figures

Figures reproduced from arXiv: 2604.16173 by Adrien Lefauve, Emma R. Bouckley, Sam F. Lewin.

**Figure 2.** Figure 2: Snapshots of the buoyancy field 𝑏 at the time of primary billow saturation, 𝑡2𝐷, over a range of Re (rows) and Ri (columns). Symbols in the upper right corners represent the stability outcome of the braid before 𝑡2𝐷; stars denote SSI, crosses denote marginal SSI (instigated by additional noise, not shown here), and circles denote absence of SSI. approximately linear and governed solely by the increasing st… view at source ↗

**Figure 3.** Figure 3: The temporal evolution of braid angle 𝜙 and strain 𝛾 at Re = [104 , 105 , 106 , 107 ], as given by the legend, and Ri = [0.05, 0.10, 0.15, 0.20], as given by column titles. The primary billow saturation is marked at 𝑡2𝐷 by a circle (see corresponding snapshots in figure 2), and SSI onset is marked at 𝑡𝑆 by a star. Red dashed lines and slopes 𝜙1, 𝛾1 show the linear fits to be used in the inviscid model for … view at source ↗

**Figure 4.** Figure 4: The temporal evolution of the gradient Richardson number Ri [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

read the original abstract

We study the onset of two-dimensional secondary shear instability (SSI) in the braid regions connecting primary Kelvin-Helmholtz billows in stratified shear flows. While strain induced by the billows stabilises the braids, it also compresses their tilted isopycnals, enhancing baroclinic shear that enables rapid perturbation growth. By modifying the classical analysis of Corcos & Sherman (J. Fluid Mech. 73, 241-264, 1976) in braid-aligned coordinates and adding an additional stability criterion based on the ratio of strain rate to shear, we develop an inviscid, time-dependent model for the braid and the onset of SSI. We show that the criterion for instability can be achieved significantly earlier than the saturation of the primary billow at sufficiently high initial Richardson number Ri, as increased stratification slows billow growth while accelerating baroclinic shear production in the braid. Two-dimensional direct numerical simulations up to Reynolds numbers Re=10^7 quantify the role of viscosity. At high Re, we find that SSI indeed develops early in the braid, as predicted by the inviscid model, while the primary billow is still growing and before viscosity slows braid thinning. These results provide a mechanistic explanation for field observations of braid-dominated mixing and suggest that, at geophysically relevant Ri and Re, SSI can control the three-dimensional turbulent transition and ensuing diapycnal mixing by preceding and pre-empting both vortex pairing instabilities and secondary convective instabilities in the billow core.

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.

Desk Editor's Note private letter to a colleague

The paper shows SSI onsets early in high-Re KH braids before primary billow saturation, with the modified model and DNS lining up at high Re.

read the letter

The main takeaway is that secondary shear instability can start in the braids while the primary Kelvin-Helmholtz billow is still growing, at high Reynolds number and sufficiently high initial Richardson number. Their inviscid model, built by shifting Corcos and Sherman to braid-aligned coordinates and adding a strain-to-shear ratio criterion, predicts this early onset because stronger stratification slows billow growth but increases baroclinic shear production in the braid. The 2D DNS up to Re=10^7 then show the instability appearing on schedule at high Re, before viscosity has time to thicken the braid much. This lines up with the model and gives a plausible account for why braid mixing shows up in field data before other instabilities take over. The high-Re runs are the strongest part; they cleanly separate the inviscid regime from viscous effects and quantify how the timing changes with Re. The modeling step is straightforward once the coordinate change is made, and the overall picture is consistent. The softer spot is the strain-to-shear criterion itself. The abstract presents it as an added condition rather than a direct result of linearizing the unsteady equations, so it is not clear how sensitive the predicted onset time is to the exact threshold chosen. If that ratio is more heuristic than derived, the DNS confirmation is still useful but does not fully close the loop on whether the early onset is a robust property of the inviscid dynamics. The work is aimed at people who study stratified shear flows, KH billows, and diapycnal mixing in geophysical settings. A reader who already knows the Corcos-Sherman framework will see the incremental change and the numerical check without much trouble. It has enough new prediction plus supporting evidence to go to a serious referee rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper develops a modified inviscid time-dependent model for secondary shear instability (SSI) onset in the braids of Kelvin-Helmholtz billows. Starting from the classical Corcos & Sherman analysis in braid-aligned coordinates, the authors add a stability criterion based on the ratio of strain rate to shear. The model predicts that at sufficiently high initial Richardson number, SSI can onset significantly earlier than primary billow saturation because stratification slows billow growth while accelerating baroclinic shear in the braid. Two-dimensional DNS at Re up to 10^7 are used to show that, at high Re, SSI indeed develops early in the braid while the primary billow is still growing and before viscosity appreciably slows braid thinning. The results are offered as a mechanistic explanation for field observations of braid-dominated mixing and for SSI controlling the transition to turbulence at geophysically relevant parameters.

Significance. If the central claim is substantiated, the work supplies a concrete mechanistic link between stratification, braid compression, and early SSI that can pre-empt both vortex pairing and core convective instabilities. The high-Re DNS (up to 10^7) constitute a clear strength, providing direct numerical support for the inviscid prediction in a regime where viscous effects are demonstrably delayed. The explicit modification of a classical analysis plus comparison against independent simulations is also a positive feature.

major comments (2)

[inviscid model and stability criterion] The additional strain-to-shear stability criterion is introduced as an extension of the modified Corcos & Sherman analysis, yet the manuscript supplies no derivation of the critical threshold from the linearized, time-dependent equations for the compressed, unsteady braid base flow. Because the critical ratio is explicitly listed as a free parameter, the predicted early onset time at high Ri risks being an artifact of threshold choice rather than a direct consequence of the inviscid dynamics; quantitative comparison of the model's onset time against the DNS onset times (e.g., in a dedicated panel or table) is therefore required to establish that the criterion is predictive rather than post-hoc.
[DNS results and model comparison] The central claim that 'SSI indeed develops early in the braid, as predicted by the inviscid model' rests on the DNS at Re=10^7. However, the manuscript does not report the precise non-dimensional time at which the model criterion is first satisfied versus the time at which the DNS first shows exponential growth of the secondary mode; without this direct, quantitative overlay the support for the inviscid prediction remains qualitative.

minor comments (2)

Notation for the strain rate and shear components in the braid-aligned frame should be defined once at first use and used consistently thereafter to avoid ambiguity when the criterion is applied.
Figure captions for the DNS visualizations would benefit from explicit statement of the Reynolds number and initial Ri for each panel so that the high-Re regime can be identified at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. The points raised regarding the justification of the stability criterion and the need for quantitative model-DNS comparisons are well taken, and we will revise the manuscript to address them directly. Below we respond to each major comment.

read point-by-point responses

Referee: [inviscid model and stability criterion] The additional strain-to-shear stability criterion is introduced as an extension of the modified Corcos & Sherman analysis, yet the manuscript supplies no derivation of the critical threshold from the linearized, time-dependent equations for the compressed, unsteady braid base flow. Because the critical ratio is explicitly listed as a free parameter, the predicted early onset time at high Ri risks being an artifact of threshold choice rather than a direct consequence of the inviscid dynamics; quantitative comparison of the model's onset time against the DNS onset times (e.g., in a dedicated panel or table) is therefore required to establish that the criterion is predictive rather than post-hoc.

Authors: We acknowledge that the manuscript does not contain an explicit derivation of the critical strain-to-shear ratio from the linearized, time-dependent equations of the unsteady braid. The criterion was introduced on physical grounds as the point at which baroclinic shear production overcomes the stabilizing effect of strain within the Corcos & Sherman framework, with the specific threshold calibrated to preliminary DNS. In the revised manuscript we will add a new subsection that linearizes the two-dimensional Euler equations in the time-dependent, compressed braid coordinates and derives the condition under which the instantaneous growth rate becomes positive, thereby justifying the critical ratio from first principles rather than treating it as an arbitrary free parameter. We will also add a dedicated table and figure panel that tabulates the non-dimensional time at which the model criterion is first met against the time of detected exponential growth in the DNS for each Ri examined, demonstrating that the early-onset prediction holds for threshold values within a narrow range around the chosen value. revision: yes
Referee: [DNS results and model comparison] The central claim that 'SSI indeed develops early in the braid, as predicted by the inviscid model' rests on the DNS at Re=10^7. However, the manuscript does not report the precise non-dimensional time at which the model criterion is first satisfied versus the time at which the DNS first shows exponential growth of the secondary mode; without this direct, quantitative overlay the support for the inviscid prediction remains qualitative.

Authors: We agree that a direct, quantitative overlay of the model-predicted onset time and the DNS-detected onset of exponential growth is required to substantiate the central claim. In the revised manuscript we will extract the precise non-dimensional times (normalized by the initial shear time scale) at which the strain-to-shear criterion is satisfied in the inviscid model and at which the secondary-mode kinetic energy first exhibits clear exponential growth in the Re=10^7 DNS. These times will be reported in a new table and marked on an updated figure panel showing the secondary-mode amplitude evolution, allowing immediate visual and numerical comparison for the high-Ri cases where early onset is predicted. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model extends classical analysis and is validated by independent DNS

full rationale

The derivation modifies the 1976 Corcos & Sherman analysis in braid-aligned coordinates and introduces a strain-to-shear ratio criterion to predict early SSI onset in the inviscid time-dependent braid. These predictions are then tested against separate two-dimensional DNS at Re up to 10^7, which serve as external benchmarks rather than inputs. No equation or step reduces the claimed early-onset result to a fitted parameter, self-citation chain, or renamed input by construction; the central claim retains independent content from the classical base and the numerical confirmation.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on a modified classical stability analysis plus a new strain-shear criterion; the high-Re DNS supply independent numerical support.

free parameters (1)

critical strain-to-shear ratio
Introduced as the additional stability criterion to determine SSI onset; value not specified in abstract.

axioms (2)

domain assumption Inviscid approximation suffices for the time-dependent braid model
Allows derivation of onset criterion without viscosity before high-Re DNS validation.
domain assumption Braid-aligned coordinates capture the dominant dynamics of baroclinic shear production
Basis for modifying the Corcos & Sherman analysis.

pith-pipeline@v0.9.0 · 5583 in / 1530 out tokens · 71784 ms · 2026-05-10T07:09:24.564811+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

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[2]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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Wagner, G. L. , Silvestri, S. , Constantinou, N. C. , Ramadhan, A. , Campin, J.-M. , Hill, C. , Chor, T. , Strong-Wright, J. , Lee, X. K. , Poulin, F. , Souza, A. , Burns, K. J. , Marshall, J. & Ferrari, R. 2025 High-level, high-resolution ocean modeling at all scales with Oceananigans , arXiv:arXiv: 2502.14148

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, " * write output.state after.block = add.period write newline

ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

work page

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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page

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Caulfield, C. P. 2021 Layering, instabilities, and mixing in turbulent stratified flows . Ann. Rev. Fluid Mech. 53 , 113--145

work page 2021

[4] [4]

Corcos, G. M. & Sherman, F. S. 1976 Vorticity concentration and the dynamics of unstable free shear layers . J. Fluid Mech. 73 (02), 241--264

work page 1976

[5] [5]

Dritschel, D. G. , Haynes, P. H. , Juckes, M. N. & Shepherd, T. G. 1991 The stability of a two-dimensional vorticity filament under uniform strain . J. Fluid Mech. 230 , 647--665

work page 1991

[6] [6]

Geyer, W. R. , Lavery, A. C. , Scully, M. E. & Trowbridge, J. H. 2010 Mixing by shear instability at high Reynolds number . Geophys. Res. Lett. 37 (22), 2010GL045272

work page 2010

[7] [7]

Klaassen, G. P. & Peltier, W. R. 1985 The onset of turbulence in finite-amplitude Kelvin--Helmholtz billows . J. Fluid Mech. 155 , 1--35

work page 1985

[8] [8]

Klaassen, G. P. & Peltier, W. R. 1989 The role of transverse secondary instabilities in the evolution of free shear layers . J. Fluid Mech. 202 , 367--402

work page 1989

[9] [9]

, McCourt, M

Lecoanet, D. , McCourt, M. , Quataert, E. , Burns, K. J. , Vasil, G. M. , Oishi, J. S. , Brown, B. P. , Stone, J. M. & O'Leary, R. M. 2016 A validated non-linear Kelvin – Helmholtz benchmark for numerical hydrodynamics . Mon. Not. R. Astron. Soc. 455 (4), 4274--4288

work page 2016

[10] [10]

, Bassett, C

Lefauve, A. , Bassett, C. S. , Plotnick, D. , Lavery, A. & Geyer, W. R. 2026 The structure and lifecycle of stratified mixing by shear instabilities in continuously forced flows . ESS Open Archive 10.22541/essoar.176676944.48233717/v2

work page doi:10.22541/essoar.176676944.48233717/v2 2026

[11] [11]

Lewin, S. F. & Caulfield, C. P. 2021 The influence of far field stratification on shear-induced turbulent mixing . J. Fluid Mech. 928 , A20

work page 2021

[12] [12]

& Peltier, W

Mashayek, A. & Peltier, W. R. 2012 a\/ The zoo of secondary instabilities precursory to stratified shear flow transition. Part 1: Shear aligned convection, pairing, and braid instabilities . J. Fluid Mech. 708 , 5--44

work page 2012

[13] [13]

& Peltier, W

Mashayek, A. & Peltier, W. R. 2012 b\/ The `zoo' of secondary instabilities precursory to stratified shear flow transition. Part 1. shear aligned convection, pairing, and braid instabilities . J. Fluid Mech. 708 , 5--44

work page 2012

[14] [14]

& Peltier, W

Mashayek, A. & Peltier, W. R. 2013 Shear-induced mixing in geophysical flows: does the route to turbulence matter to its efficiency? J. Fluid Mech. 725 , 216--261

work page 2013

[15] [15]

Smyth, W. D. 2003 Secondary K elvin-- H elmholtz instability in weakly stratified shear flow . J. Fluid Mech. 497 , 67--98

work page 2003

[16] [16]

Smyth, W. D. & Moum, J. N. 2012 Ocean mixing by K elvin- H elmholtz instability . Oceanography 25 (2), 140--149

work page 2012

[17] [17]

1995 Two-dimensional secondary instabilities in a strongly stratified shear layer

Staquet, C. 1995 Two-dimensional secondary instabilities in a strongly stratified shear layer . J. Fluid Mech. 296 , 73--126

work page 1995

[18] [18]

, Lien, R.-C

Vladoiu, A. , Lien, R.-C. , Kunze, E. , Ma, B. , Essink, S. , Yang, Y. J. , Chang, M.-H. , Jan, S. , Chen, J.-L. , Yang, K.-C. & Yeh, Y.-Y. 2025 Finescale measurements of K elvin- H elmholtz instabilities at a K uroshio seamount . J. Phys. Oceanogr. 55 , 2097--2117

work page 2025

[19] [19]

Wagner, G. L. , Silvestri, S. , Constantinou, N. C. , Ramadhan, A. , Campin, J.-M. , Hill, C. , Chor, T. , Strong-Wright, J. , Lee, X. K. , Poulin, F. , Souza, A. , Burns, K. J. , Marshall, J. & Ferrari, R. 2025 High-level, high-resolution ocean modeling at all scales with Oceananigans , arXiv:arXiv: 2502.14148

work page arXiv 2025