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Rotational flow underlying coupled surface and internal waves. I: Eulerian perspective

David Henry, Rossen I. Ivanov, Zisis N. Sakellaris

TL;DR

This paper analyzes coupled surface and internal gravity waves in a two-layer fluid with an upper rotational layer of constant vorticity $\gamma_1\neq0$ and a lower irrotational layer. It develops a phase-plane (Eulerian) framework from a linear wave ansatz via a Hamiltonian formulation, deriving a dispersion relation expressed as a quartic $P_4(c)=0$ and identifying how vorticity and a key parameter $A$ shape flow topologies. The main contribution is a comprehensive classification of qualitative streamline patterns across three regimes of $|A|<1$, $A=1$, and $A>1$, including the presence or absence of critical layers and the emergence of cat's-eye–like vortical structures, all in the moving frame. The results illuminate wave–current interactions in rotating two-layer systems and set the stage for subsequent Lagrangian analyses of particle motion in these flows.

Abstract

In this paper we examine the flow generated by coupled surface and internal small-amplitude water waves in a two-fluid layer model, where we take the upper layer to be rotational (constant vorticity) and the lower layer to be irrotational. The presence of vorticity greatly complicates the underlying analysis, yet it generates a rich array of otherwise unobservable phenomena such as the presence of critical layers, and stagnation points, in the fluid interior. We employ a phase-plane analysis to elucidate the qualitative behaviour of streamlines for a range of different coupled-wave, and vorticity, regimes. Although the water waves considered are linear in the fluid dynamics sense, the dynamical systems which govern their motion are nonlinear.

Rotational flow underlying coupled surface and internal waves. I: Eulerian perspective

TL;DR

This paper analyzes coupled surface and internal gravity waves in a two-layer fluid with an upper rotational layer of constant vorticity and a lower irrotational layer. It develops a phase-plane (Eulerian) framework from a linear wave ansatz via a Hamiltonian formulation, deriving a dispersion relation expressed as a quartic and identifying how vorticity and a key parameter shape flow topologies. The main contribution is a comprehensive classification of qualitative streamline patterns across three regimes of , , and , including the presence or absence of critical layers and the emergence of cat's-eye–like vortical structures, all in the moving frame. The results illuminate wave–current interactions in rotating two-layer systems and set the stage for subsequent Lagrangian analyses of particle motion in these flows.

Abstract

In this paper we examine the flow generated by coupled surface and internal small-amplitude water waves in a two-fluid layer model, where we take the upper layer to be rotational (constant vorticity) and the lower layer to be irrotational. The presence of vorticity greatly complicates the underlying analysis, yet it generates a rich array of otherwise unobservable phenomena such as the presence of critical layers, and stagnation points, in the fluid interior. We employ a phase-plane analysis to elucidate the qualitative behaviour of streamlines for a range of different coupled-wave, and vorticity, regimes. Although the water waves considered are linear in the fluid dynamics sense, the dynamical systems which govern their motion are nonlinear.
Paper Structure (19 sections, 10 theorems, 179 equations, 15 figures)

This paper contains 19 sections, 10 theorems, 179 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Equations of motion
  3. Linear approximation of the equations
  4. Hamiltonian formulation
  5. Wave solutions
  6. Dispersion relation
  7. Phase plane analysis
  8. Lower fluid layer
  9. Upper fluid layer
  10. The case $\mathbf{A=1}$
  11. $A=1$ and $\mathfrak{c}>0$ (no critical layers)
  12. $A=1$ and $\mathfrak{c}<0$ (critical layer present)
  13. The case $A<1$
  14. $|A|<1$ and $\mathfrak{c}>0$ (no critical layers)
  15. $|A|<1$, and $\mathfrak{c}<0$ (critical layer present)
  16. ...and 4 more sections

Key Result

Lemma 3.1

Figures (15)

  • Figure 1: Coupled surface-internal water waves for $a/a_1>0$, and $\gamma_1<0$.
  • Figure 2: Coupled surface-internal water waves for $a/a_1<0$, and $\gamma_1>0$.
  • Figure 3: Phase portrait for the lower-fluid layer. The dotted grey line represents the $\infty-$isocline, with the dotted-dashed lines representing the $0-$isoclines. The internal wave profile () with mean-water level $Y=kh$ (corresponding to $y=0$) is also illustrated.
  • Figure 4: The two real-valued branches of the Lambert $W$ function.
  • Figure 5: Phase portrait for $A=1$, $\gamma_1>0$, $\mathfrak{c}>0$, and where \ref{['Cond:A=1-a']} holds. The dotted grey lines represent the $\infty-$isoclines, while the dotted-dashed lines represent the $0-$isoclines. The surface wave profile () has mean-water level $Y_1=0$, corresponding to $y=h_1$. The internal wave profile () has mean water level $Y=kh_1$, corresponding to $y=0$. Note that the streamline representing the surface wave lies beneath that of the internal wave in terms of the $(X,Y_1)-$coordinates used for this phase portrait.
  • ...and 10 more figures

Theorems & Definitions (41)

  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Remark 3.8
  • ...and 31 more