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arxiv: 2606.12627 · v1 · pith:H3EMYBMUnew · submitted 2026-06-10 · ⚛️ physics.flu-dyn · astro-ph.SR

Two pathways to diapycnal mixing in strongly stratified flows with no initial vertical shear

Pascale Garaud , Dante Buhl , Jason Johnstone , Arstanbek Tulekeyev , Nathan van Duker This is my paper

Pith reviewed 2026-06-27 07:55 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn astro-ph.SR
keywords stratified flowshorizontal shear instabilitydiapycnal mixingKelvin-Helmholtz instabilitybuoyancy Reynolds numbervertical shear emergenceturbulence pathways
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The pith

Horizontal shear instabilities in strongly stratified flows inevitably generate vertical shear that triggers small-scale Kelvin-Helmholtz instabilities at large buoyancy Reynolds number.

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

The paper studies turbulence and diapycnal mixing in flows with strong vertical stratification but shear only in the horizontal direction and no initial vertical shear. Linear theory and direct numerical simulations reveal two pathways by which primary horizontal shear instabilities produce vertical shear. In both pathways the resulting vertical shear becomes unstable to small-scale Kelvin-Helmholtz instabilities once the buoyancy Reynolds number is large enough, leading to mixing. The pathways differ in the vertical scales they excite and therefore reach different peak mixing efficiencies. A reader would care because this demonstrates that mixing can arise spontaneously even when the mean Richardson number is large or infinite, provided the Reynolds number is high.

Core claim

In low Froude number, high Reynolds number horizontally-sheared stratified flows with no initial vertical shear, vertical shear emerges either directly from vertically-modulated eigenmodes of the primary instability or indirectly via nonlinear evolution into long-lived columnar vortices followed by three-dimensional hyperbolic instabilities; this vertical shear then drives secondary or tertiary small-scale Kelvin-Helmholtz instabilities on the buoyancy scale at sufficiently large buoyancy Reynolds number Re_b, producing diapycnal mixing whose efficiency depends on which pathway is taken.

What carries the argument

Spontaneous emergence of vertical shear as a by-product of horizontal shear instabilities, which then triggers small-scale Kelvin-Helmholtz instabilities on the buoyancy scale once Re_b is large.

If this is right

  • Diapycnal mixing occurs in strongly stratified flows even without any initial vertical shear once the buoyancy Reynolds number is sufficiently large.
  • The two pathways excite different ranges of vertical scales and therefore produce different peak mixing efficiencies.
  • Horizontal shear instabilities remain active and can drive turbulence even when the mean Richardson number is large or formally infinite.
  • The final step in both pathways is the onset of small-scale Kelvin-Helmholtz instabilities on the buoyancy scale.
  • A vertically-invariant primary mode can first produce a long-lived time-dependent two-dimensional vortical flow before three-dimensional instabilities generate vertical shear.

Where Pith is reading between the lines

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

  • These pathways may account for observed mixing in oceanic regions where horizontal shear from currents or eddies is present but vertical shear is weak.
  • The scale selection difference between pathways could guide the development of mixing parameterizations that depend on the dominant horizontal shear geometry.
  • Similar sequences might appear in other high-Reynolds-number stratified systems such as atmospheric jets or stellar interiors when horizontal shear is the only available energy source.
  • Extending the simulations to include background rotation or variable stratification could reveal whether the pathways remain robust or are suppressed.

Load-bearing premise

The direct numerical simulations fully resolve the secondary and tertiary instabilities without significant numerical dissipation or domain-size effects altering the emergence of vertical shear or the measured mixing efficiencies.

What would settle it

A well-resolved simulation at large Re_b in which vertical shear fails to appear from the horizontal shear instability or in which the observed mixing efficiency matches neither of the two described pathways.

Figures

Figures reproduced from arXiv: 2606.12627 by Arstanbek Tulekeyev, Dante Buhl, Jason Johnstone, Nathan van Duker, Pascale Garaud.

Figure 1
Figure 1. Figure 1: Illustration of the two possible pathways from a vertically-invariant horizontal shear flow, to the development of small-scale KH instabilities of the emergent vertical shear. This assumes that 𝐹𝑟 ≪ 1 (to ensure that 3D primary modes are unstable) and 𝑅𝑒𝑏 ≫ 1 (to ensure that KH instabilities on small scales can develop). The top row illustrates the pathway P1 described for instance in the body-forced DNS o… view at source ↗
Figure 2
Figure 2. Figure 2: Results from the set of DNS1 described in §2.1, with 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.15 (solid lines, DNS1a) and 𝐹𝑟 = 0.1 (dashed lines, DNS1b). (a) Rms velocities 𝑢rms (orange), 𝑣rms (green) and 𝑤rms (black), and buoyancy fluctuation 𝑏rms (red). The black dotted line shows the predicted growth of the fastest-growing mode of primary instability (with growth rate 𝜆 = 0.26). (b) Area fraction of the 𝑦 = 0 plane (lin… view at source ↗
Figure 3
Figure 3. Figure 3: Snapshots of the vertical velocity field 𝑤 for DNS1a (𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.15), initialized with white noise. (a) During the exponential growth phase of the primary mode of instability at 𝑡 = 56. (b) During the saturation of the primary mode of instability at 𝑡 = 78. 0 X0-7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results from the set of DNS2 described in §2.2, with 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.15 (solid lines, DNS2a) and 𝐹𝑟 = 0.1 (dashed lines, DNS2b). (a) Rms velocities 𝑢rms (orange), 𝑣rms (green) and 𝑤rms (black), and buoyancy fluctuation 𝑏rms (red). The green dashed line shows the predicted growth of the corresponding fastest-growing mode of primary instability (with growth rate 𝜆 = 0.26), and the black dotted lines … view at source ↗
Figure 5
Figure 5. Figure 5: Snapshots of the vertical velocity field 𝑤 for DNS2a (𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.15), initialized with a seed meander for the 𝑘 𝑧 = 0 mode. (a) During the exponential growth phase of the secondary mode of instability at 𝑡 = 85. (b) During the saturation of the secondary mode of instability at 𝑡 = 125. 0 X0-10 Rapids articles must not exceed this page length [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Contour lines of the stream function 𝜓b representing the meandering flow ub, overlaid on top of color maps of |ub | (see equation 3.1), for increasing meander amplitude 𝑎 from panel (a) to (d). Note that the color range is different in the top row and bottom row, and that the colors of the streamlines are arbitrary. that ub exhibits meanders whose amplitude increases with 𝑎. For 𝑎 ≥ |𝑣 −1 0 |/2 = 0.55, the… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Growth rate of the secondary mode of instability as a function of 𝑘 𝑧 , for 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.1, and varying 𝑎 (see legend for detail). In each case 𝑀 = 20, 𝑁 = 10. (b) As in (a), but fixing 𝑎 = 1 and 𝐹𝑟 = 0.08, and varying 𝑅𝑒 (with 𝑃𝑒 = 𝑅𝑒). In both panels, we have checked that 𝜆 has converged given this Fourier resolution (see Appendix B for detail). of the 𝜆(𝑘 𝑧) curve downwards and towards lo… view at source ↗
Figure 8
Figure 8. Figure 8: (a) Growth rate of the secondary instability 𝜆(𝑘 𝑧 ) for 𝑅𝑒 = 𝑃𝑒 = 10000, 𝑎 = 1, 𝑀 = 20, and varying 𝐹𝑟 (see legend). (b) Wavenumber 𝑘max at which 𝜆(𝑘 𝑧 ) peaks, as a function of 𝐹𝑟, for 𝑅𝑒 = 𝑃𝑒 = 10000 and three different values of 𝑎 (see legend). The dashed line is the line 𝑘max = 𝐹𝑟−1 . (c) Growth rate 𝜆max = max𝑘𝑧 𝜆(𝑘 𝑧 ) as a function of 𝐹𝑟, for the same parameters as panel (b). types of hyperbolic in… view at source ↗
Figure 9
Figure 9. Figure 9: Left column: The background vorticity field 𝜔b is shown as a color map, and the background streamfunction 𝜓b is overlaid as a contour map. The values of the meander amplitude 𝑎 used for each map is shown to the left of the panel. The corresponding structure of the eigenmode for 𝜔 ′ 𝑧 and 𝑤 ′ , computed for 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.1 are shown in the center and right columns, respectively. The value of 𝑘 𝑧 u… view at source ↗
Figure 10
Figure 10. Figure 10: Symbols show the growth rate 𝜆max of the fastest-growing mode of secondary instability as a function of meander amplitude 𝑎, for 𝑅𝑒 = 𝑃𝑒 = 10000 and different values of 𝐹𝑟 (see legend). Also shown is the growth rate of the fastest-growing mode of the primary instability (𝜆 = 0.26, green dashed line), the growth rate of the hyperbolic instability for an isolated hyperbolic point 𝛥h (blue line, see equation… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between the results of DNS2b (𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.1), in the left column, with linear theory for the secondary instability of the steady flow ub,DNS given in equation (3.17) with 𝐴 = 0.4, 𝑉0 = −0.93, 𝑈0 = 0.045 in the right column. (a,c) DNS snapshots of 𝜔𝑧 and 𝑤, respectively, at 𝑡 = 84 in the 𝑧 = 0 plane. (b) 𝛥 2 (see equation 3.15) for the same snapshot. (d) Snapshot of 𝑤 at the same tim… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of various quantities associated with energy dissipation in DNS1a (purple lines and symbols) and DNS2a (green lines and symbols), where both have 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.15). (a) ⟨|∇u| 2 ⟩ (𝑡) against effective buoyancy Reynolds number 𝑅𝑒𝑏,eff (𝑡) defined in (4.1), starting from 𝑡 = 𝑡1/4, the time when the area fraction of the 𝑦 = 0 plane occupied by regions with 𝑅𝑖 < 1/4 first begins to exceed… view at source ↗
Figure 13
Figure 13. Figure 13: (a) Real part of the growth rate, 𝜆, as a function of 𝑘 𝑥 , for 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.1, and 𝑘 𝑧 = 0, obtained using different Fourier resolutions (see legend). (b) Colormap of ˜𝜔𝑧 for the 𝑘 𝑥 = 0.5 mode, which is the fastest-growing mode in a domain of length 𝐿𝑥 = 4𝜋. The parameters are the same as in (a), with 𝑁 = 10. (c) Comparison of ˜𝜔𝑧 (𝑥, 3𝜋/2), for the 𝑁 = 1 and 𝑁 = 10 resolutions. The parameter… view at source ↗
Figure 14
Figure 14. Figure 14: Analysis of the effect of the selected Fourier resolution 𝑀 (with 𝑁 = 𝑀/2) on the mode growth rate, for 𝑅𝑒 = 𝑃𝑒 = 10000, 𝑘 𝑥 = 0.5, 𝐹𝑟 = 0.1 and 𝑎 = 1. (a) Comparison of 𝜆(𝑘 𝑧 ) obtained with different resolutions, shown in the legend. The black arrow shows the largest value of 𝑘 𝑧 for which we estimate that the growth rate 𝜆 is accurately computed using 𝑀 = 20, 𝑁 = 10, by comparison with 𝑀 = 16, 𝑁 = 8. (… view at source ↗
Figure 15
Figure 15. Figure 15: Visualizations of the vertical velocity 𝑤 ′ (panels (a) and (c)) and of the vertical vorticity 𝜔 ′ 𝑧 (panels (b) and (d)) of the fastest-growing mode of instability of ub for 𝑅𝑒 = 𝑃𝑒 = 10000, 𝐹𝑟 = 0.1, 𝑎 = 1 and 𝑘 𝑧 = 8. The top row (panels (a) and (b)) shows a ’low’ resolution solution with 𝑀 = 20, 𝑁 = 10 and the bottom row (panels (c) and (d)) shows a higher-resolution one with 𝑀 = 40, 𝑁 = 20. Note that… view at source ↗
read the original abstract

While vertically-sheared stratified flows have been studied extensively, their horizontally-sheared counterparts have received considerably less attention. Yet, horizontal shear instabilities remain active even when the mean Richardson number is large or even formally infinite, and can drive turbulence in strongly stratified (low Froude number) flows at sufficiently high Reynolds number. In this work, we combine linear theory with direct numerical simulations to investigate two pathways to turbulence in low Froude / high Reynolds number horizontally-sheared flow with no initial vertical shear. In the first pathway, vertical shear emerges directly from vertically-modulated eigenmodes of the primary horizontal shear instability, and becomes unstable to secondary small-scale Kelvin-Helmholtz (KH) instabilities on the buoyancy scale at sufficiently large buoyancy Reynolds number $Re_b$. In the second pathway, a vertically-invariant eigenmode of the primary horizontal shear instability initially dominates, causing the background flow to evolve nonlinearly into a long-lived time-dependent two-dimensional (columnar) vortical flow. The vortices are subsequently unstable to secondary three-dimensional hyperbolic instabilities from which vertical shear emerges, which is finally unstable to tertiary small-scale KH instabilities on the buoyancy scale at sufficiently large $Re_b$. This shows that the emergence of vertical shear driving small-scale KH instabilities is an inevitable by-product of horizontal shear instabilities in strongly stratified flows at sufficiently large $Re_b$. However, we also find that the two pathways excite different ranges of vertical scales, which results in different peak mixing efficiencies.

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 / 1 minor

Summary. The manuscript combines linear stability theory with direct numerical simulations to examine mixing in horizontally sheared, strongly stratified flows (low Froude number) that have no initial vertical shear. It identifies two pathways to diapycnal mixing: (1) vertically modulated eigenmodes of the primary horizontal shear instability that directly generate vertical shear, which then becomes unstable to small-scale Kelvin-Helmholtz (KH) instabilities once the buoyancy Reynolds number Re_b is sufficiently large; (2) a vertically invariant eigenmode that evolves into long-lived columnar vortices, which undergo secondary hyperbolic instabilities that generate vertical shear, followed by tertiary KH instabilities at large Re_b. The central claim is that emergence of vertical shear (and subsequent KH mixing) is an inevitable byproduct of horizontal shear instabilities at large enough Re_b, although the pathways excite different vertical scales and therefore produce different peak mixing efficiencies.

Significance. If the numerical results are robust, the work clarifies how horizontal shear alone can sustain diapycnal mixing in strongly stratified regimes relevant to geophysical flows, without requiring initial vertical shear. The distinction between the two pathways and their differing efficiencies supplies a concrete, falsifiable prediction that could be tested in future simulations or experiments. The integration of linear theory to interpret the DNS evolution is a methodological strength.

major comments (2)
  1. [DNS section] DNS section (implicit in the abstract and linear-theory discussion): the headline claim that vertical shear and buoyancy-scale KH instabilities emerge 'inevitably' at large Re_b rests on the simulations faithfully capturing the secondary and tertiary instabilities. No explicit resolution criteria (e.g., grid points per buoyancy length, verification that local Ri < 1/4 is resolved without excessive numerical dissipation, or checks against domain-height artifacts) are reported. Insufficient vertical resolution or domain size could suppress the small-scale KH or artificially constrain the vertical scales, directly affecting the reported difference in peak mixing efficiencies between pathways.
  2. [Results on the two pathways] Results on the two pathways: the assertion that the pathways excite 'different ranges of vertical scales' and therefore different peak mixing efficiencies is load-bearing for the paper's novelty. Without tabulated quantitative diagnostics (e.g., spectra of vertical shear or mixing efficiency versus Re_b for each pathway) or convergence tests with respect to domain height, it remains unclear whether the efficiency contrast is physical or an artifact of the chosen vertical domain.
minor comments (1)
  1. [Abstract] The abstract states that the second pathway produces a 'long-lived time-dependent two-dimensional (columnar) vortical flow,' but the transition criterion from the primary eigenmode to this state is not quantified; a brief statement of the relevant time scale or amplitude threshold would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive comments on the DNS methodology and quantitative support for the two pathways. We address each major comment below and will revise the manuscript to incorporate the requested details and diagnostics.

read point-by-point responses

  1. Referee: [DNS section] DNS section (implicit in the abstract and linear-theory discussion): the headline claim that vertical shear and buoyancy-scale KH instabilities emerge 'inevitably' at large Re_b rests on the simulations faithfully capturing the secondary and tertiary instabilities. No explicit resolution criteria (e.g., grid points per buoyancy length, verification that local Ri < 1/4 is resolved without excessive numerical dissipation, or checks against domain-height artifacts) are reported. Insufficient vertical resolution or domain size could suppress the small-scale KH or artificially constrain the vertical scales, directly affecting the reported difference in peak mixing efficiencies between pathways.

    Authors: We agree that explicit documentation of resolution criteria is necessary to substantiate the claims of inevitability at large Re_b. In the revised manuscript we will add a dedicated paragraph (or subsection) reporting: (i) grid points per buoyancy length (typically >10 in the vertical), (ii) verification that local gradient Richardson numbers below 1/4 are adequately resolved without excessive numerical dissipation, and (iii) domain-height sensitivity tests confirming that the observed pathways and efficiency differences persist when the vertical domain is doubled. These checks were performed during the original study and will now be reported explicitly. revision: yes

  2. Referee: [Results on the two pathways] Results on the two pathways: the assertion that the pathways excite 'different ranges of vertical scales' and therefore different peak mixing efficiencies is load-bearing for the paper's novelty. Without tabulated quantitative diagnostics (e.g., spectra of vertical shear or mixing efficiency versus Re_b for each pathway) or convergence tests with respect to domain height, it remains unclear whether the efficiency contrast is physical or an artifact of the chosen vertical domain.

    Authors: The linear stability analysis already quantifies the distinct vertical wavenumbers of the two eigenmodes, and the DNS spectra reflect the subsequent nonlinear evolution. To make the efficiency contrast fully quantitative and demonstrably robust, the revision will include: (i) tabulated spectra of vertical shear for representative cases of each pathway, (ii) mixing efficiency plotted versus Re_b for both pathways, and (iii) explicit convergence tests with respect to vertical domain height showing that the peak efficiencies remain distinct. These additions will be placed in a new figure or table and accompanying text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results emerge from linear eigenmode analysis plus DNS of the governing equations

full rationale

The paper derives its central claims (two distinct pathways to vertical shear and KH mixing) directly from linear stability analysis of the horizontally sheared base flow followed by DNS evolution of the Navier-Stokes-Boussinesq equations at low Fr/high Re_b. No parameters are fitted to data and then relabeled as predictions; no uniqueness theorems or ansatzes are imported via self-citation; the reported differences in vertical scales and mixing efficiencies are outputs of the simulations rather than inputs. The derivation chain is therefore self-contained against the external benchmark of the governing PDEs and does not reduce to its own assumptions by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Work rests on the incompressible Navier-Stokes equations under the Boussinesq approximation for density stratification; no new physical entities are introduced. The buoyancy Reynolds number threshold is treated as an input parameter rather than derived.

free parameters (1)
  • buoyancy Reynolds number threshold
    The condition 'sufficiently large Re_b' for KH onset is stated but not derived from first principles within the paper.
axioms (2)
  • domain assumption Boussinesq approximation holds for the density variations considered
    Standard modeling choice for stratified flows at the Froude numbers studied.
  • standard math Navier-Stokes equations accurately describe the fluid motion
    Fundamental governing equations invoked throughout.

pith-pipeline@v0.9.1-grok · 5815 in / 1425 out tokens · 16534 ms · 2026-06-27T07:55:02.485136+00:00 · methodology

discussion (0)

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