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Benjamin-Feir Instability of Interfacial Gravity-Capillary Waves in a Two-Layer Fluid. Part II. Surface-Tension Effects

Olga Avramenko, Volodymyr Naradovyi

TL;DR

This work extends Part I by treating surface tension $T$ as an independent control parameter in a two-layer inviscid fluid, and derives a nonlinear Schrödinger reduction together with long‑wave asymptotics to map modulational stability in $(\rho,k)$ as $T$ varies. By constructing three-dimensional critical surfaces $J=0$, $J\to\infty$, and $\omega''=0$ in $(\rho,k,T)$, the authors reveal loop, corridor, and cut stability structures whose planar sections reproduce the stability maps, and they identify the Bond threshold $T^* = h_1^2/3$ and the golden-ratio depth ratio $h_2/h_1 = \varphi$ as key organizing features. The analysis shows how the balance between gravity and capillarity, redistributed by depth asymmetry, governs the nonlinear coefficient $J$ and the dispersion curvature, leading to topology changes in modulational stability as $T$ crosses $T^*$ or as $h_2/h_1$ crosses the golden ratio. The resulting framework provides a complete geometric classification of stability regimes and offers a foundation for extensions involving shear, currents, or additional interfaces.

Abstract

This second part of the study develops a geometric and asymptotic description of how surface tension governs the modulational stability of interfacial waves in a two-layer fluid. Extending the analytical framework of Part~I, surface tension is treated as a freely adjustable parameter, enabling one to trace how nonlinear and dispersive properties evolve across a broad range of depth ratios and density contrasts. Using the nonlinear Schrödinger reduction together with long-wave asymptotics, the mechanisms shaping the boundaries between stable and unstable regimes are identified and their dependence on surface tension is quantified. The long-wave structure is governed by two density values that determine the bases of the loop and the corridor on the stability diagrams. Their ordering switches only when the lower layer is deeper, and loop-type structures arise exclusively in this regime. A second organising parameter is the classical Bond threshold, at which the dispersive and nonlinear singularities coincide. When surface tension exceeds this value and the upper layer is sufficiently deep, resonant and dispersive effects generate a capillary cut that replaces the corridor and characterises strongly capillary, upper-layer-dominated configurations. To unify these observations, full three-dimensional critical surfaces separating different types of nonlinear and dispersive behaviour are computed. The loop, corridor, and cut arise as planar sections of these surfaces, and their transitions follow from the deformation of the intersection between the resonant and dispersive sheets. Two depth ratios correspond to geometric degeneracies: equal layer depths, where the intersection reduces to a straight line, and the golden-ratio configuration, where the critical surface becomes horizontally tangent at the Bond threshold.

Benjamin-Feir Instability of Interfacial Gravity-Capillary Waves in a Two-Layer Fluid. Part II. Surface-Tension Effects

TL;DR

This work extends Part I by treating surface tension as an independent control parameter in a two-layer inviscid fluid, and derives a nonlinear Schrödinger reduction together with long‑wave asymptotics to map modulational stability in as varies. By constructing three-dimensional critical surfaces , , and in , the authors reveal loop, corridor, and cut stability structures whose planar sections reproduce the stability maps, and they identify the Bond threshold and the golden-ratio depth ratio as key organizing features. The analysis shows how the balance between gravity and capillarity, redistributed by depth asymmetry, governs the nonlinear coefficient and the dispersion curvature, leading to topology changes in modulational stability as crosses or as crosses the golden ratio. The resulting framework provides a complete geometric classification of stability regimes and offers a foundation for extensions involving shear, currents, or additional interfaces.

Abstract

This second part of the study develops a geometric and asymptotic description of how surface tension governs the modulational stability of interfacial waves in a two-layer fluid. Extending the analytical framework of Part~I, surface tension is treated as a freely adjustable parameter, enabling one to trace how nonlinear and dispersive properties evolve across a broad range of depth ratios and density contrasts. Using the nonlinear Schrödinger reduction together with long-wave asymptotics, the mechanisms shaping the boundaries between stable and unstable regimes are identified and their dependence on surface tension is quantified. The long-wave structure is governed by two density values that determine the bases of the loop and the corridor on the stability diagrams. Their ordering switches only when the lower layer is deeper, and loop-type structures arise exclusively in this regime. A second organising parameter is the classical Bond threshold, at which the dispersive and nonlinear singularities coincide. When surface tension exceeds this value and the upper layer is sufficiently deep, resonant and dispersive effects generate a capillary cut that replaces the corridor and characterises strongly capillary, upper-layer-dominated configurations. To unify these observations, full three-dimensional critical surfaces separating different types of nonlinear and dispersive behaviour are computed. The loop, corridor, and cut arise as planar sections of these surfaces, and their transitions follow from the deformation of the intersection between the resonant and dispersive sheets. Two depth ratios correspond to geometric degeneracies: equal layer depths, where the intersection reduces to a straight line, and the golden-ratio configuration, where the critical surface becomes horizontally tangent at the Bond threshold.

Paper Structure

This paper contains 18 sections, 27 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem formulation, preliminary background, and limiting cases
  3. Mathematical formulation and background assumptions
  4. Limiting case $\rho=1$ for equal layer depths
  5. Mutual placement of the points $\rho_{\mathrm L}$ and $\rho_{\mathrm C}$ at $k=0$
  6. Long-wave critical surface tension $T^{\ast}$ (Bond threshold $\mathrm{Bo}=1/3$)
  7. Geometric condition for horizontal tangency of the critical surface
  8. Stability diagrams: the influence of surface tension
  9. General remarks and reference configuration
  10. Symmetric and asymmetric configurations of the upper unstable region
  11. Corridor and cut formation
  12. Loop and its interaction with the corridor
  13. Summary and physical interpretation
  14. Three-dimensional critical surfaces $T(\rho,k)$
  15. Analytical framework and Bond-threshold structure
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Modulational–stability diagrams for all combinations of $h_{1}, h_{2} \in \{1, 5, 9, 13\}$ for $T=1/2, 1, 2$.
  • Figure 2: The cut-type corridor for $h_1=5$ and $h_2\in\{50,100,150\}$ for $T=1,\,25/3,\,20,\,40,\,60$.
  • Figure 3: Stability diagrams for surface–tension coefficients $T=1/2,\,3/5,\,1,\,5/3,\,2$ for $h_1=5$ and $h_2\in\{1,2,3,4,4.75,5,5.25,6\}$.
  • Figure 4: Critical surfaces $T(\rho,k)$ for $h_1=5$ and $h_2\in\{1,4,5,6,5(1+\sqrt{5})/{2},12,50,100,150\}$.