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On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet

Banavara N. Shashikanth

TL;DR

This work develops an intrinsic Hamiltonian framework for the evolution of a closed vortex sheet in a plane, separating two immiscible, inviscid fluids. By decomposing the problem into two Zakharov-type flows in the outer and inner domains, the authors construct a boundary Poisson bracket on the sheet that yields a compact, intrinsic set of evolution equations for the sheet position and its strength, including density-jump generalizations with density-weighted variables. The paper derives explicit expressions for the vortex-sheet bracket and the Hamiltonian (kinetic energy, plus potential energy in the jump case), computes functional derivatives, and identifies Lagrangian invariants under the Cauchy principal value transport, drawing connections to Kelvin’s circulation theorem and a KdV-like bracket. The intrinsic formulation avoids domain-boundary operators, offering potential advantages for structure-preserving numerical schemes and insights into two-fluid interface dynamics, with future directions including extensions to vortex patches and more general interfaces. Overall, the work provides a rigorous Hamiltonian perspective on co-dimension-1 vortex dynamics and highlights deep connections to established Hamiltonian structures in fluid mechanics.

Abstract

Motivated by the work of previous authors on vortex sheets and their applications, the intrinsic inviscid evolution equations of a closed vortex sheet in a plane, separating two piecewise constant density fluids, and their Hamiltonian form are investigated. The model has potential applications to problems involving the dynamics of interfaces of two immiscible fluids. A boundary Poisson bracket, which appears to be new and related to the KdV bracket, is obtained containing the curve-tangential derivative $\partial / \partial s$. Lagrangian invariants of the sheet motion by its self-induced velocity--the Cauchy principal value of the Biot-Savart integral--are also derived.

On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet

TL;DR

This work develops an intrinsic Hamiltonian framework for the evolution of a closed vortex sheet in a plane, separating two immiscible, inviscid fluids. By decomposing the problem into two Zakharov-type flows in the outer and inner domains, the authors construct a boundary Poisson bracket on the sheet that yields a compact, intrinsic set of evolution equations for the sheet position and its strength, including density-jump generalizations with density-weighted variables. The paper derives explicit expressions for the vortex-sheet bracket and the Hamiltonian (kinetic energy, plus potential energy in the jump case), computes functional derivatives, and identifies Lagrangian invariants under the Cauchy principal value transport, drawing connections to Kelvin’s circulation theorem and a KdV-like bracket. The intrinsic formulation avoids domain-boundary operators, offering potential advantages for structure-preserving numerical schemes and insights into two-fluid interface dynamics, with future directions including extensions to vortex patches and more general interfaces. Overall, the work provides a rigorous Hamiltonian perspective on co-dimension-1 vortex dynamics and highlights deep connections to established Hamiltonian structures in fluid mechanics.

Abstract

Motivated by the work of previous authors on vortex sheets and their applications, the intrinsic inviscid evolution equations of a closed vortex sheet in a plane, separating two piecewise constant density fluids, and their Hamiltonian form are investigated. The model has potential applications to problems involving the dynamics of interfaces of two immiscible fluids. A boundary Poisson bracket, which appears to be new and related to the KdV bracket, is obtained containing the curve-tangential derivative . Lagrangian invariants of the sheet motion by its self-induced velocity--the Cauchy principal value of the Biot-Savart integral--are also derived.
Paper Structure (22 sections, 3 theorems, 99 equations, 2 figures)

This paper contains 22 sections, 3 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Zakharov's problem and vortex sheets.
  3. Hamiltonian structure for a closed vortex sheet via Zakharov's canonical bracket.
  4. The case of a uniform density field.
  5. Vortex sheet bracket.
  6. Hamiltonian function and functional derivatives.
  7. Hamiltonian vector field.
  8. Remarks.
  9. The case of a density field with a jump.
  10. Vortex sheet bracket.
  11. Hamiltonian function and functional derivatives.
  12. Remarks.
  13. Obtaining the $\tilde{\gamma}$ equation in (\ref{['eq:evol2']}) from Euler's equation.
  14. Remark.
  15. Lagrangian invariants of the sheet motion.
  16. ...and 7 more sections

Key Result

Proposition 1

The evolution of a closed vortex sheet in a homogeneous fluid is governed by the PDE system (eq:evol1) combined with (eq:cpv), (eq:cpvo) and (eq:cpvi). Together they form a system of intrinsic equations for the evolution of the closed vortex sheet. This is a Hamiltonian system of the form (eq:hameqn

Figures (2)

  • Figure 1: Schematic sketch of the 3D free surface problem considered by Zakharov
  • Figure 2: Sketch illustrating the computation of the $\partial \hat{t}/\partial t_{\mid_S}$ term.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3