
Hamiltonian structure of the
KelvinHelmholtz instability

 

This twopart paper reports on various new insights into the classic
KelvinHelmholtz problem which models the instability of a plane
vortex sheet
and the complicated motions arising therefrom. The full nonlinear
version of the hydrodynamic problem is treated, with allowance for
gravity and surface tension, and the account deals in precise fashion
with several inherently peculiar properties of the mathematical
model. The main achievement of Part 1, presented in Section 3, is to
demonstrate that the problem admits a canonical Hamiltonian
formulation, which represents a novel variational definition of a
functional representing perturbations in kinetic energy. The
Hamiltonian structure thus revealed is then used to account
systematically for relations between symmetries and conservation laws,
and none of those examined appears to have been noticed before. In
Section 4, a generalized, noncanonical Hamiltonian structure is shown
to apply when the vortex sheet becomes folded, so requiring a
parametric representation, as is well known to occur in the later
stages of evolution from KelvinHelmholtz instability. Further
invariant properties are demonstrated in this context. Finally,
Section 5, the linearized version of the problem  reviewed briefly in
Section 2.1  is reappraised in the light of Hamiltonian structure, and it is shown how KelvinHelmholtz instability can be interpreted as the coincidence of wave modes characterized respectively by positive and negative values of the Hamiltonian functional representing perturbations in total energy.

 

T.B. Benjamin & T.J. Bridges.
Reappraisal of the KelvinHelmholtz instability. Part 1: Hamiltonian
structure,
J. Fluid Mech. 333 301325 (1997)
JFM website


T.B. Benjamin & T.J. Bridges.
Reappraisal of the KelvinHelmholtz instability. Part 2: Interaction of the KelvinHelmholtz, superharmonic and BenjaminFeir instabilities,
J. Fluid Mech. 333 327373 (1997)
JFM website


