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Nonlinear Rayleigh-Taylor instability of the viscous surface wave in an infinitely deep ocean

Tien-Tai Nguyen

TL;DR

This work analyzes a viscous incompressible, nonhomogeneous fluid with density increasing with depth, bounded above by a moving free surface in a vertical gravity field. By a Beale-type Lagrangian transformation, the authors derive a fixed-domain formulation and perform a detailed linear spectral analysis around the hydrostatic equilibrium, proving the existence of infinitely many real growing normal modes and identifying a maximal linear growth rate $\\Lambda$. Building on a Gaardner-type nonlinear energy framework (Guo–Strauss and Grenier), the paper extends to nonlinear RT instability by constructing a wide class of initial data from linear modes and obtaining sharp a priori energy estimates that control the nonlinear perturbations. The results establish nonlinear instability in the viscous, density-stratified setting without surface tension, with a rigorous link between linear growth and nonlinear growth through carefully crafted initial data and robust energy methods, highlighting the potential for RT-driven growth in geophysical and engineering contexts. The analysis combines spectral theory, variational methods, and intricate energy estimates in a fixed-domain, periodically bounded, infinitely deep geometry.

Abstract

In this paper, we consider an incompressible viscous fluid in an infinitely deep ocean, being bounded above by a free moving boundary. The governing equations are the gravity-driven incompressible Navier-Stokes equations with variable density and no surface tension is taken into account on the free surface. After using the Lagrangian transformation, we write the main equations in a perturbed form in a fixed domain. In the first part, we describe a spectral analysis of the linearized equations around a hydrostatic equilibrium $(ρ_0(x_3), 0, P_0(x_3))$ for a smooth increasing density profile $ρ_0$. Precisely, we prove that there exist infinitely many normal modes to the linearized equations by following the operator method initiated by Lafitte and Nguyen. In the second part, we study the nonlinear Rayleigh-Taylor instability around the above profile by constructing a \textit{wide class} of initial data for the nonlinear perturbation problem departing from the equilibrium, based on the finding of infinitely many normal modes. Our nonlinear result follows the previous framework of Guo and Strauss and also of Grenier with a refinement.

Nonlinear Rayleigh-Taylor instability of the viscous surface wave in an infinitely deep ocean

TL;DR

This work analyzes a viscous incompressible, nonhomogeneous fluid with density increasing with depth, bounded above by a moving free surface in a vertical gravity field. By a Beale-type Lagrangian transformation, the authors derive a fixed-domain formulation and perform a detailed linear spectral analysis around the hydrostatic equilibrium, proving the existence of infinitely many real growing normal modes and identifying a maximal linear growth rate . Building on a Gaardner-type nonlinear energy framework (Guo–Strauss and Grenier), the paper extends to nonlinear RT instability by constructing a wide class of initial data from linear modes and obtaining sharp a priori energy estimates that control the nonlinear perturbations. The results establish nonlinear instability in the viscous, density-stratified setting without surface tension, with a rigorous link between linear growth and nonlinear growth through carefully crafted initial data and robust energy methods, highlighting the potential for RT-driven growth in geophysical and engineering contexts. The analysis combines spectral theory, variational methods, and intricate energy estimates in a fixed-domain, periodically bounded, infinitely deep geometry.

Abstract

In this paper, we consider an incompressible viscous fluid in an infinitely deep ocean, being bounded above by a free moving boundary. The governing equations are the gravity-driven incompressible Navier-Stokes equations with variable density and no surface tension is taken into account on the free surface. After using the Lagrangian transformation, we write the main equations in a perturbed form in a fixed domain. In the first part, we describe a spectral analysis of the linearized equations around a hydrostatic equilibrium for a smooth increasing density profile . Precisely, we prove that there exist infinitely many normal modes to the linearized equations by following the operator method initiated by Lafitte and Nguyen. In the second part, we study the nonlinear Rayleigh-Taylor instability around the above profile by constructing a \textit{wide class} of initial data for the nonlinear perturbation problem departing from the equilibrium, based on the finding of infinitely many normal modes. Our nonlinear result follows the previous framework of Guo and Strauss and also of Grenier with a refinement.
Paper Structure (24 sections, 41 theorems, 594 equations)

This paper contains 24 sections, 41 theorems, 594 equations.

Table of Contents

  1. Introduction
  2. The governing equations and main results
  3. Formulation in Lagrangian coordinates
  4. Equilibrium state and and main results
  5. The linear analysis
  6. Solutions on the outer region $(-\infty,-a)$ and reduction to an ODE on the finite interval $(-a,0)$
  7. A bilinear form and a self-adjoint invertible operator
  8. Proof of Theorem \ref{['ThmLinear']}
  9. Maximal growth rate
  10. A priori energy estimates
  11. Energy estimates of the perturbation transport
  12. Temporal estimates for the perturbation velocity
  13. Horizontal estimates of the perturbation velocity
  14. Estimates of the perturbation density
  15. Elliptic estimates
  16. ...and 9 more sections

Key Result

Lemma 2.1

For any $\mathbf k \in (L^{-1}\mathbb Z)^2\setminus \{0\}$,

Theorems & Definitions (73)

  • Lemma 2.1
  • Theorem 2.1
  • Theorem 2.2
  • proof : Proof of Lemma \ref{['LemEigenvalueReal']}
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.2
  • ...and 63 more