pith. sign in

arxiv: 2605.00335 · v1 · submitted 2026-05-01 · ⚛️ physics.flu-dyn

Dynamics of finger-type convection in double-diffusive instability

Mohammad Mohaghar , Anirban Bhattacharjee , Suhas S. Jain , Donald R. Webster This is my paper

Pith reviewed 2026-05-09 19:15 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords double-diffusive instabilityfinger-type convectionbuoyancy anomalyfingertip trackingvortex ringscalar mixinglaboratory experimentdirect numerical simulation
0
0 comments X

The pith

Buoyancy anomaly evolution links finger-type convection growth phases to a time-dependent force balance.

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

Finger-type convection in double-diffusive instability proceeds through three distinct stages of fingertip motion: acceleration, quasi-steady propagation, and decay. The paper shows that these stages arise directly from the time evolution of the buoyancy anomaly, which first increases to drive acceleration, balances against shear resistance to sustain steady speed, and then diminishes through dilution and top-boundary effects to cause slowdown. This mechanism is quantified using synchronized PLIF and PIV measurements in a sealed laboratory facility across three salinity contrasts, matched with high-resolution DNS, and processed through a systematic fingertip detection and tracking framework. The approach establishes consistency between experiment, simulation, and linear stability analysis while documenting how fingertip vortex rings remain symmetric at moderate salinity but become asymmetric at higher values, shifting transport from vertical to lateral. The link matters because finger convection governs scalar mixing rates in many stratified flows.

Core claim

In experiments and matched DNS at fixed thermal contrast and salinity contrasts of 350, 450, and 550 ppm, fingertip growth curves display three phases whose peak rates increase with salinity contrast while nondimensional height histories collapse. The evolution of the buoyancy anomaly supplies the connection to a time-dependent force balance: rising buoyancy produces acceleration, shear-induced resistance maintains quasi-steady propagation, and progressive dilution together with boundary influence produces late deceleration. At 450 ppm the fingertips develop symmetric vortex rings that organize vertical transport; at 550 ppm stronger buoyancy destabilizes the rings into asymmetric roll-up, z

What carries the argument

The evolving buoyancy anomaly at the fingertips, which supplies the time-dependent force balance that sequences acceleration, quasi-steady propagation, and decay.

Load-bearing premise

The sealed-surface laboratory facility with the chosen salinity contrasts accurately represents the intermediate regime of finger-type double-diffusive instability without dominant boundary artifacts or detection bias from the fingertip tracking framework.

What would settle it

Direct time series of the buoyancy anomaly measured at tracked fingertips that fail to show the predicted increase during acceleration, balance during quasi-steady propagation, and decrease during decay would falsify the claimed link between anomaly evolution and growth-rate phases.

Figures

Figures reproduced from arXiv: 2605.00335 by Anirban Bhattacharjee, Donald R. Webster, Mohammad Mohaghar, Suhas S. Jain.

Figure 1
Figure 1. Figure 1: Schematic of the experimental setup. (a) Perspective view showing the overall setup. (b) Close-up of the tank and FEP injection tube with holes for controlled release. meter. The procedure was repeated for each experiment to achieve the prescribed salinity contrasts. A thin acrylic lid was placed on top of the tank to create a sealed-surface configuration. This lid suppressed free-surface deformation and e… view at source ↗
Figure 2
Figure 2. Figure 2: Example of a velocity–scalar–vorticity field for the case with 𝛥𝑆 = 450 ppm at 𝑡 = 240 s. The velocity vectors (green) are overlaid on the vorticity field (red–blue colormap) and the concentration field (grayscale colormap). (a) Full field of view showing the spatial distribution of vortical structures and scalar gradients, with the red box indicating the zoom region. (b) Zoomed-in view of the selected reg… view at source ↗
Figure 3
Figure 3. Figure 3: Example sequential processing steps for automated fingertip detection: (1) concentration field; (2) diffuser region masked; (3) Canny edge detection applied at multiple intensity thresholds (three thresholds shown) to identify fingertip edges; (4) individual tip masks formed from closed-loop contours matching the fingertip geometry; (5) combined fingertip mask created by merging all fingertip masks for var… view at source ↗
Figure 4
Figure 4. Figure 4: Demonstration of the fingertip detection method for three salinity contrasts at the same experimental time: (a) 𝛥𝑆 = 350 ppm, (b) 𝛥𝑆 = 450 ppm, and (c) 𝛥𝑆 = 550 ppm. In each case, the detected fingertip contours (green) are overlaid on the corrected concentration field at 𝑡 = 240 s, illustrating the method’s capability to identify fingertip locations across a range of salinity contrasts. primary tool for i… view at source ↗
Figure 5
Figure 5. Figure 5: Three-dimensional simulation configuration and grid convergence assessment. (a) Computational domain showing the initial scalar field with hot, salty fluid (red) overlying cold, fresh fluid (blue). (b) Iso￾surface of the initial interface perturbation. (c) Grid-sensitivity study for the 450 ppm case showing the temporal evolution of finger-tip height ℎ(𝑡) for three grid resolutions: 64 × 256 × 32, 128 × 51… view at source ↗
Figure 6
Figure 6. Figure 6: Experimental finger growth analysis across salinity contrasts: (a) trajectories of all detected fingertips for the 𝛥𝑆 = 450 ppm case (as an example), where ℎ is the finger height and the trajectories are additionally colored as a function of time-of-initiation. (b) Ensemble-averaged fingertip height ⟨ℎ − ℎ0⟩ versus time (𝑡 − 𝑡0) for 𝛥𝑆 = 350, 450, and 550 ppm, with shaded ±1 standard deviation bands; 𝑡0 an… view at source ↗
Figure 7
Figure 7. Figure 7: Computational finger growth analysis across salinity contrasts: (a) time evolution of finger height ℎ versus time 𝑡 for salinity contrasts of 350, 450, and 550 ppm. (b) Non-dimensionalized representation of the same data (finger height is normalized by the maximum height ℎmax and time is scaled by the characteristic growth timescale 𝑡𝑐). 3.1.2. Computational growth rate and validation view at source ↗
Figure 8
Figure 8. Figure 8: Temporal evolution of mixed-material area and net salt transport for three salinity contrasts for the experimental results. (a,b) Mixed-material area. (c,d) Area-averaged net salt transport. J𝑆,𝑖(𝑡) = 1 𝑉 ∫ 𝑉 |𝐶𝑆𝑢𝑖 | d𝑉, J𝑇,𝑖(𝑡) = 1 𝑉 ∫ 𝑉 |𝐶𝑇𝑢𝑖 | d𝑉, (3.13) where 𝑖 ∈ {𝑥, 𝑦, 𝑧}. Thus, the experimental transport measures represent planar area averages, whereas the DNS results correspond to full three-dimensi… view at source ↗
Figure 9
Figure 9. Figure 9: The experimental concentration field 𝐶 (1st column), scalar dissipation rate 𝜒 (2nd column), horizontal salt flux 𝐶𝑢𝑥 (3rd column), vertical salt flux 𝐶𝑢𝑦 (4th column) and divergence of salt flux ∇· (𝐶𝒖) (5th column). Data are shown for nondimensional time 𝑡 ∗ = 1.02 for three salinity contrasts: 𝛥𝑆 = 350 ppm (top row), 450 ppm (middle row), and 550 ppm (bottom row). 0 X0-21 view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the computational scalar fields and flux components at the same nondimensional time 𝑡 ∗ = 0.75. Rows correspond to salinity contrast: 𝛥𝑆 = 350 ppm (top row), 450 ppm (middle row), and 550 ppm (bottom row). From left to right, the columns show three-dimensional volume renderings of the salinity field 𝑆, the temperature field 𝑇, the vertical temperature flux 𝐶𝑇 𝑢𝑦, and the salt flux components… view at source ↗
Figure 11
Figure 11. Figure 11: Temporal evolution of the volume-averaged scalar flux components from the simulations for different salinity contrasts: (a,b) horizontal salt flux, J𝑆,𝑥 ; (c,d) vertical salt flux, J𝑆,𝑦; (e,f) out-of-plane salt flux, J𝑆,𝑧 ; and (g,h) vertical temperature flux, J𝑇,𝑦. Results are shown for 𝛥𝑆 = 350, 450, and 550 ppm. 0 X0-24 view at source ↗
Figure 12
Figure 12. Figure 12: Temporal evolution of a single salt finger for the 𝛥𝑆 = 450 ppm experimental case, tracked from its initial formation through growth, deformation, and eventual breakdown and mixing. Each panel shows mean-subtracted velocity vectors (green) overlaid on the vorticity field (red–blue colormap) and the concentration field (grayscale colormap). The time point and the subtracted average velocity are indicated a… view at source ↗
Figure 13
Figure 13. Figure 13: Temporal evolution of the scalar dissipation rate for a representative salt finger for the 𝛥𝑆 = 450 ppm experimental case (the same salt finger shown in figure 12), covering the finger’s development from initial formation to breakdown and mixing. During the intermediate steady-elongation stage (middle row), finger growth proceeds with an approximately constant upward propagation rate, consistent with the … view at source ↗
Figure 14
Figure 14. Figure 14: Temporal evolution of the vertical salt flux field, 𝐶𝑢𝑦, for a representative salt finger for the 𝛥𝑆 = 450 ppm experimental case (the same salt finger shown in figure 12), covering its development from initial formation through mature growth and eventual breakdown and mixing. the cap emerges and the neck forms. A localized region of weakly elevated vertical flux is observed at the same location, indicatin… view at source ↗
Figure 15
Figure 15. Figure 15: Temporal evolution of integrated quantities for a representative salt finger for the 𝛥𝑆 = 450 ppm experimental case, showing its evolution from formation to breakdown: (a) circulation for the counter-clockwise rotating fluid (blue dashed line) and clockwise rotating fluid (red dash-dotted line) and the total amount (black line). (b) Area-averaged enstrophy. (c) Area-averaged scalar dissipation rate. (d) A… view at source ↗
Figure 16
Figure 16. Figure 16: Example of a representative salt finger for the 𝛥𝑆 = 550 ppm experimental case at 𝑡 = 160 s. (a) Full concentration field showing the finger formation, with the red box highlighting the zoom region. (b) Concentration field for the zoomed-in region, capturing the head of the finger structure. (c) Scalar dissipation rate field, 𝜒, for the zoomed-in region. 3.4. Finger-scale dynamics: zig-zag and lateral-dri… view at source ↗
Figure 17
Figure 17. Figure 17: Temporal evolution of the scalar dissipation rate field, 𝜒, for representative salt fingers for the 𝛥𝑆 = 550 ppm experimental case. cores beneath each evolving tip, similar to the lower-salinity case. However, these vertical flux cores now appear tilted with significant horizontal displacement. In corroboration, the sequence of horizontal flux fields (figure 20) reveals elevated regions of alternating pos… view at source ↗
Figure 18
Figure 18. Figure 18: Vertically-averaged profiles of scalar dissipation rate 𝜒 𝑦 at two time points for a representative finger for (a) 𝛥𝑆 = 450 ppm and (b) 𝛥𝑆 = 550 ppm experimental cases. The two profiles in each panel are separated by the same nondimensional time interval, 𝛥𝑡∗ = 0.074. Insets show the corresponding 𝜒 fields at the same time points (the field with the red border corresponds to the red profile; the field wit… view at source ↗
Figure 19
Figure 19. Figure 19: Temporal evolution of the vertical salt flux field, 𝐶𝑢𝑦, for representative salt fingers for the 𝛥𝑆 = 550 ppm experimental case (same time sequence as shown in figure 17). profile is removed, 𝑏 ′ (𝑥, 𝑦, 𝑧, 𝑡) = 𝑏(𝑥, 𝑦, 𝑧, 𝑡) − 𝑏 𝑥,𝑧 (𝑦, 𝑡), (3.17) where the overbar denotes averaging in the horizontal (𝑥, 𝑧) directions for a fixed value of 𝑦. The perturbation field 𝑏 ′ therefore represents buoyancy anomali… view at source ↗
Figure 20
Figure 20. Figure 20: Temporal evolution of the horizontal salt flux field, 𝐶𝑢𝑥 , for representative salt fingers for the 𝛥𝑆 = 550 ppm experimental case (same time sequence as shown in figure 17). The sequence shows the fingers’ development from the early stage through growth, deformation, and eventual breakdown. figure 21, in which regions of elevated 𝑏 ′ 𝑇 remain confined to a thin region near the interface, whereas the 𝑏 ′ … view at source ↗
Figure 21
Figure 21. Figure 21: Three-dimensional volume renderings of the buoyancy 𝑏 and its decomposition at representative times in the simulations. From left to right: the total buoyancy field 𝑏 at 𝑡 = 150 s for the 𝛥𝑆 = 450 ppm case; the horizontally averaged (𝑥–𝑧) buoyancy field 𝑏 𝑥𝑧 at 𝑡 = 0 s; the temperature-dominated buoyancy anomaly 𝑏 ′ 𝑇 at 𝑡 = 150 s; the salinity-dominated buoyancy anomaly 𝑏 ′ 𝑆 at 𝑡 = 150 s; and the total … view at source ↗
Figure 22
Figure 22. Figure 22: (a) Three-dimensional volume renderings of the buoyancy anomaly 𝑏 ′ at selected time points during the development of finger-type convection for the 𝛥𝑆 = 450 ppm case in the simulation, with the maximum and minimum of the fingertip buoyancy anomaly 𝑏 ′ tip indicated for each time. (b) Temporal evolution of the corresponding extrema, showing 𝑏 ′ tip (max) and − 𝑏 ′ tip (min) as functions of time. smooths t… view at source ↗
read the original abstract

Finger-type convection in double-diffusive instability (DDI) controls mixing and scalar transport in many stratified flows, yet a quantitative, finger-resolved description of the transient growth, transport, and saturation pathways has been limited. Here, finger-type DDI is analyzed in a sealed-surface laboratory facility using synchronized planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) at fixed thermal contrast $\Delta T=5^\circ$C and three salinity contrasts, $\Delta S=350$, 450, and 550 ppm, complemented by a matched high-resolution three-dimensional DNS. A systematic fingertip detection and tracking framework generates ensemble growth curves. Fingertip growth follows a sequence of three stages (acceleration, quasi-steady propagation, and decay). The peak growth rates increase monotonically with $\Delta S$, and nondimensional fingertip-height histories collapse onto a common trend. The peak growth rates are reproduced by DNS and agree with linear stability analysis, establishing experiment--DNS--theory consistency in the intermediate regime. The mixed-material area increases with time, initially following a common nondimensional trend before transitioning to $\Delta S$-dependent interaction and breakdown. Finger-scale measurements reveal the formation of a symmetric vortex ring at the fingertips for $\Delta S=450$ ppm, inducing vertical-aligned transport. At $\Delta S=550$ ppm the roll-up becomes asymmetric: stronger buoyancy amplifies shear, destabilizes the vortex ring, and produces a zig-zag/lateral-drift mode that enhances the lateral transport. Finally, the evolution of the buoyancy anomaly links the growth-rate phases to a time-dependent force balance in which increasing buoyancy drives acceleration, shear-induced resistance regulates quasi-steady propagation, and dilution with top-boundary influence yields late-stage fingertip deceleration.

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 examines finger-type double-diffusive instability (DDI) in a sealed laboratory facility using synchronized PLIF/PIV measurements at fixed ΔT = 5°C and three salinity contrasts (ΔS = 350, 450, 550 ppm), supported by matched 3D DNS. It reports three distinct stages of fingertip growth (acceleration, quasi-steady propagation, decay), monotonic increase of peak growth rates with ΔS, collapse of nondimensional fingertip-height histories, agreement of peak rates with linear stability analysis, formation of symmetric or asymmetric vortex rings depending on ΔS, and an interpretation of the buoyancy-anomaly evolution as establishing a time-dependent force balance (buoyancy-driven acceleration, shear-regulated propagation, and late-stage deceleration from dilution plus top-boundary effects). Mixed-material area growth and lateral transport modes are also quantified.

Significance. If the reported experiment–DNS–theory consistency and the buoyancy-anomaly linkage to force-balance regimes hold, the work supplies a quantitative, finger-resolved description of transient growth and saturation pathways that is currently limited in the DDI literature. The ensemble tracking, vortex-ring observations, and nondimensional collapse are useful for refining mixing parameterizations in stratified flows. The direct comparison to existing linear stability analysis and the use of high-resolution DNS constitute clear strengths.

major comments (2)
  1. [abstract (buoyancy anomaly discussion) and corresponding results section on growth phases] The central claim that the evolution of the buoyancy anomaly establishes three distinct force-balance regimes (abstract, final paragraph) rests on the late-stage deceleration being caused by top-boundary influence and dilution. No domain-height variation, comparison to periodic-boundary DNS, or sensitivity test is described, leaving open the possibility that the observed decay phase and the inferred force balance reflect the sealed-surface facility rather than intrinsic finger dynamics. This is load-bearing because the three-stage interpretation and the experiment–DNS consistency are asserted for the intermediate regime.
  2. [methods section describing the PLIF/PIV tracking framework] The fingertip detection and tracking framework that generates the ensemble growth curves and buoyancy-anomaly time series is not validated against independent methods (e.g., manual annotation or alternative image-processing thresholds). Systematic bias in timing or amplitude of the detected peaks would directly affect the extracted force-balance phases and the reported collapse of nondimensional histories.
minor comments (2)
  1. [experimental parameters] Units for salinity contrast are given as ppm; explicit conversion to practical salinity units or density difference would aid reproducibility.
  2. [DNS methods] The manuscript would benefit from a brief statement on the grid resolution and domain aspect ratio used in the matched DNS to allow direct comparison with the laboratory facility dimensions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and positive assessment of the significance of our work. We address each of the major comments point by point below.

read point-by-point responses

  1. Referee: [abstract (buoyancy anomaly discussion) and corresponding results section on growth phases] The central claim that the evolution of the buoyancy anomaly establishes three distinct force-balance regimes (abstract, final paragraph) rests on the late-stage deceleration being caused by top-boundary influence and dilution. No domain-height variation, comparison to periodic-boundary DNS, or sensitivity test is described, leaving open the possibility that the observed decay phase and the inferred force balance reflect the sealed-surface facility rather than intrinsic finger dynamics. This is load-bearing because the three-stage interpretation and the experiment–DNS consistency are asserted for the intermediate regime.

    Authors: We agree that the absence of domain-height variations or periodic-boundary simulations leaves the attribution of the late-stage decay open to interpretation. The DNS domain is matched to the experimental facility, including the top boundary condition, and the decay phase is observed consistently in both. The primary support for the three stages comes from the buoyancy anomaly time series and the matching of peak growth rates to linear theory in the intermediate regime. In the revised manuscript, we will expand the discussion to explicitly note this limitation and emphasize that the acceleration and quasi-steady phases are robustly supported by the multi-method agreement, while the decay phase may include boundary effects. We will also clarify that the force-balance interpretation is observational rather than definitive. revision: partial

  2. Referee: [methods section describing the PLIF/PIV tracking framework] The fingertip detection and tracking framework that generates the ensemble growth curves and buoyancy-anomaly time series is not validated against independent methods (e.g., manual annotation or alternative image-processing thresholds). Systematic bias in timing or amplitude of the detected peaks would directly affect the extracted force-balance phases and the reported collapse of nondimensional histories.

    Authors: We acknowledge the importance of validating the automated tracking method. In the revised version, we will add a validation subsection in the methods, where we compare the automated fingertip detection results against manual tracking on a representative sample of images from each ΔS case. This will include quantitative metrics on the agreement of detected peak times and growth rates to demonstrate that any bias is minimal and does not affect the reported trends. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on direct measurements, independent DNS, and external linear stability analysis.

full rationale

The paper's core results—fingertip growth stages, peak growth rates, mixed-material area evolution, vortex structures, and buoyancy-anomaly force balance—are obtained from direct PLIF/PIV tracking in the sealed facility, ensemble averaging, and high-resolution DNS. Peak growth rates are compared to (not derived from) independent linear stability analysis. The time-dependent force balance is an interpretive link extracted from observed buoyancy anomaly histories rather than a fitted parameter or self-referential definition. No self-citations are invoked as load-bearing uniqueness theorems, no ansatzes are smuggled, and no predictions reduce by construction to inputs defined by the same dataset. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claims rest on direct experimental measurements at fixed laboratory conditions, matched DNS, and comparison to pre-existing linear stability analysis. No new free parameters are introduced or fitted, and no new physical entities are postulated.

pith-pipeline@v0.9.0 · 5631 in / 1302 out tokens · 52051 ms · 2026-05-09T19:15:50.259957+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages

  1. [1]

    1980 , booktitle =

    Nonlinear Evolution and Structure of Salt Fingers , editor =. 1980 , booktitle =. doi:https://doi.org/10.1016/S0422-9894(08)71222-4 , author =

    work page doi:10.1016/s0422-9894(08)71222-4 1980

  2. [2]

    Experimental investigation of non-linear standing internal waves using combined density and velocity measurements , author=. Exp. Fluids , volume=. 2023 , publisher=

    work page 2023

  3. [3]

    , title =

    Shen, C.Y. , title =. Phys. Fluids. , volume =. 1995 , month =

    work page 1995

  4. [4]

    The evolution of the double-diffusive instability: Salt fingers , author=. Phys. Fluids A , volume=. 1989 , publisher=

    work page 1989

  5. [5]

    Deep-Sea Res

    An ocean curiosity: the perpetual salt fountain , author=. Deep-Sea Res. , volume=

    work page

  6. [6]

    salt-fountain

    The “salt-fountain” and thermohaline convection , author=. Tellus , volume=. 1960 , publisher=

    work page 1960

  7. [7]

    Double-diffusive phenomena , author=. Annu. Rev. Fluid Mech. , volume=. 1974 , publisher=

    work page 1974

  8. [8]

    Double diffusion in oceanography , author=. Annu. Rev. Fluid Mech. , volume=

    work page

  9. [9]

    2013 , publisher=

    Double-diffusive convection , author=. 2013 , publisher=

    work page 2013

  10. [10]

    The characteristics of salt fingers in a variety of fluid systems, including stellar interiors, liquid metals, oceans, and magmas , author=. Phys. Fluids , volume=. 1983 , publisher=

    work page 1983

  11. [11]

    Deep-Sea Res

    The growth rate of super-critical salt fingers , author=. Deep-Sea Res. A , volume=. 1979 , publisher=

    work page 1979

  12. [12]

    The coupled turbulent transports of salt and heat across a sharp density interface , author=. Int. J. Heat Mass Transf. , volume=. 1965 , publisher=

    work page 1965

  13. [13]

    Experiments on double diffusive convection , author=. C. R. Phys. , volume=

    work page

  14. [14]

    A new case of convection in the presence of combined vertical salinity and temperature gradients , author=. Proc. Natl. Acad. Sci. U.S.A. , volume=

    work page

  15. [15]

    Collective instability of salt fingers , author=. J. Fluid Mech. , volume=. 1969 , publisher=

    work page 1969

  16. [16]

    Deep-Sea Res

    Salt fingers and convecting layers , author=. Deep-Sea Res. Oceanogr. Abstr. , volume=. 1969 , organization=

    work page 1969

  17. [17]

    Deep-Sea Res

    Laboratory experiments on the structure of salt fingers , author=. Deep-Sea Res. A , volume=. 1989 , publisher=

    work page 1989

  18. [18]

    Anisotropy of salt fingers , author=. J. Phys. Oceanogr. , volume=

    work page

  19. [19]

    Observations of the cell structure of salt fingers , author=. J. Fluid Mech. , volume=. 1970 , publisher=

    work page 1970

  20. [20]

    Flux measurements across a finger interface at low values of the stability ratio , author=. J. Mar. Res. , volume=

    work page

  21. [21]

    Double-diffusive convection with imposed vertical mass flux , author=. J. Mar. Res. , volume=

    work page

  22. [22]

    Double-diffusive interleaving on horizontal gradients , author=. J. Fluid Mech. , volume=. 2006 , publisher=

    work page 2006

  23. [23]

    High Rayleigh number convection with double diffusive fingers , author=. Phys. Fluids , volume=. 2010 , publisher=

    work page 2010

  24. [24]

    Transition to finger convection in double-diffusive convection , author=. Phys. Fluids , volume=. 2014 , publisher=

    work page 2014

  25. [25]

    Origin of thermohaline staircases , author=. J. Phys. Oceanogr. , volume=

    work page

  26. [26]

    Astrophys

    Turbulent mixing and layer formation in double-diffusive convection: Three-dimensional numerical simulations and theory , author=. Astrophys. J. , volume=. 2011 , publisher=

    work page 2011

  27. [27]

    Scaling laws and flow structures of double diffusive convection in the finger regime , author=. J. Fluid Mech. , volume=. 2016 , publisher=

    work page 2016

  28. [28]

    From convection rolls to finger convection in double-diffusive turbulence , author=. Proc. Natl. Acad. Sci. U.S.A. , volume=. 2016 , publisher=

    work page 2016

  29. [29]

    Multiple states and transport properties of double-diffusive convection turbulence , author=. Proc. Natl. Acad. Sci. U.S.A. , volume=. 2020 , publisher=

    work page 2020

  30. [30]

    Numerical experiments on salt-finger convection , author=. Prog. Oceanogr. , volume=. 2003 , publisher=

    work page 2003

  31. [31]

    Direct numerical simulation of salt sheets and turbulence in a double-diffusive shear layer , author=. Geophys. Res. Lett. , volume=. 2007 , publisher=

    work page 2007

  32. [32]

    Copepod behavior responses around internal waves , author=. Front. Mar. Sci. , volume=. 2020 , publisher=

    work page 2020

  33. [33]

    2019 , type =

    Effects of initial conditions and Mach number on turbulent mixing transition of shock-driven variable-density flow , author =. 2019 , type =

    work page 2019

  34. [34]

    Evaluation of turbulent mixing transition in a shock-driven variable-density flow , author=. J. Fluid Mech. , volume=. 2017 , publisher=

    work page 2017

  35. [35]

    A review of oceanic salt-fingering theory , author=. Prog. Oceanogr. , volume=. 2003 , publisher=

    work page 2003

  36. [36]

    Form of the temperature-salinity relationship in the central water: Evidence for double-diffusive mixing , author=. J. Phys. Oceanogr. , volume=

    work page

  37. [37]

    A new vision of giant planet interiors: Impact of double diffusive convection , author=. Astron. Astrophys. , volume=. 2012 , publisher=

    work page 2012

  38. [38]

    Layered convection as the origin of Saturn’s luminosity anomaly , author=. Nat. Geosci. , volume=. 2013 , publisher=

    work page 2013

  39. [39]

    Astrophys

    Chemical transport and spontaneous layer formation in fingering convection in astrophysics , author=. Astrophys. J. , volume=. 2013 , publisher=

    work page 2013

  40. [40]

    Astrophys

    Numerical simulations of thermohaline convection: implications for extra-mixing in low-mass RGB stars , author=. Astrophys. J. , volume=. 2010 , publisher=

    work page 2010

  41. [41]

    On finite amplitude instability in thermohaline convection , author=. J. Mar. Res. , volume=

    work page

  42. [42]

    On thermohaline convection with linear gradients , author=. J. Fluid Mech. , volume=. 1969 , publisher=

    work page 1969

  43. [43]

    Deep-Sea Res

    On the structure of salt fingers , author=. Deep-Sea Res. Oceanogr. Abstr. , volume=. 1973 , organization=

    work page 1973

  44. [44]

    On the vertical transport due to fingers in double diffusive convection , author=. J. Fluid Mech. , volume=. 1972 , publisher=

    work page 1972

  45. [45]

    Experiments on double-diffusive sugar--salt fingers at high stability ratio , author=. J. Fluid Mech. , volume=. 1996 , publisher=

    work page 1996

  46. [46]

    Double-diffusive finger convection: influence of concentration at fixed buoyancy ratio , author=. J. Fluid Mech. , volume=. 2002 , publisher=

    work page 2002

  47. [47]

    A mechanism for layer formation in a double-diffusive fluid , author=. J. Fluid Mech. , volume=. 2003 , publisher=

    work page 2003

  48. [48]

    Staircase formation in unstably stratified double diffusive finger convection , author=. Phys. Fluids , volume=. 2022 , publisher=

    work page 2022

  49. [49]

    Double-diffusive transport in laboratory thermohaline staircases , author=. J. Fluid Mech. , volume=. 2003 , publisher=

    work page 2003

  50. [50]

    Stability analysis , author=

    Double-diffusive instability in an inclined fluid layer Part 2. Stability analysis , author=. J. Fluid Mech. , volume=. 1980 , publisher=

    work page 1980

  51. [51]

    Double-diffusive instability in an inclined fluid layer. Part 1. Experimental investigation , author=. J. Fluid Mech. , volume=. 1980 , publisher=

    work page 1980

  52. [52]

    Water Resour

    Experimental investigation of the stability boundary for double-diffusive finger convection in a Hele-Shaw cell , author=. Water Resour. Res. , volume=. 1997 , publisher=

    work page 1997

  53. [53]

    On the collective instability of salt fingers , author=. J. Fluid Mech. , volume=. 1981 , publisher=

    work page 1981

  54. [54]

    Astrophys

    Double-diffusive mixing in stellar interiors in the presence of horizontal gradients , author=. Astrophys. J. Lett. , volume=. 2014 , publisher=

    work page 2014

  55. [55]

    Astrophys

    2D or not 2D: the effect of dimensionality on the dynamics of fingering convection at low Prandtl number , author=. Astrophys. J. , volume=. 2015 , publisher=

    work page 2015

  56. [56]

    Double-diffusive transport in multicomponent vertical convection , author=. Phys. Rev. Fluids , volume=. 2023 , publisher=

    work page 2023

  57. [57]

    Double-diffusive convection at low Prandtl number , author=. Annu. Rev. Fluid Mech. , volume=. 2018 , publisher=

    work page 2018

  58. [58]

    Accurate conservative phase-field method for simulation of two-phase flows , author=. J. Comput. Phys. , volume=. 2022 , publisher=

    work page 2022

  59. [59]

    A model for transport of interface-confined scalars and insoluble surfactants in two-phase flows , author=. J. Comput. Phys. , volume=. 2024 , publisher=

    work page 2024

  60. [60]

    Stationary states of forced two-phase turbulence , author=. Chem. Eng. J. , volume=. 2025 , publisher=

    work page 2025

  61. [61]

    arXiv preprint arXiv:2510.10857 , year=

    A model for transport of soluble surfactants in two-phase flows , author=. arXiv preprint arXiv:2510.10857 , year=

    work page arXiv

  62. [62]

    Vortex rings from cylinders with inclined exits , author=. Phys. Fluids , volume=. 1998 , publisher=

    work page 1998