Normal forms for wave motion in fluid interfaces

Abstract

The subject of this paper is the dynamics of wave motion in the two-dimensional Kelvin-Helmholtz problem for an interface between two immiscible fluids of different densities. The difference of the mean flow between the two fluid bodies is taken to be zero, and the effects of surface tension are neglected. We transform the problem to Birkhoff normal form, in which a precise analysis can be made of classes of resonant solutions. This paper studies standing-wave solutions of the fourth-order normal form in particular detail. We find that that there are families of invariant resonant subsystems, which are nevertheless integrable. Within these families we describe the periodic and the time quasi-periodic standing waves, and determine their stability or instability. In particular we show that for a certain range of densities, a basic time-periodic standing wave with principal wave number k is unstable to modes with principal wave numbers k/4 and 9k/4, and we calculate the Lyapunov exponent of the instability. We furthermore show that the stable and unstable manifolds to these periodic solutions of the Birkhoff normal form are connected by a homoclinic orbit. This instability mechanism, as well as others that we describe, appears to be new, and its description is possible because of the precision afforded by the normal form. These results contrast with the case of the water wave problem described by Dyachenko & Zakharov [1] and Craig & Worfolk [2], where the fourth-order Birkhoff normal form is an integrable system, with all orbits undergoing stable almost-periodic motion, and instabilities arise only in normal forms to higher order.

Deep-water internal solitary waves near critical density ratio

“…A Hamiltonian formulation of the problem of a free interface between two ideal fluids, under rigid lid boundary conditions for the upper fluid, was also given by Benjamin & Bridges [21]. Craig & Groves [22] give a similar expression, by using the Dirichlet-Neumann operators for both the upper and lower fluid domains (see also [23]). The Hamiltonian can be expanded in series with respect to powers of the canonical variables.…”

Boundary Perturbation Methods for Water Waves

“…Thus, for a simulation in d = 2 dimensions using N x collocation points and N perturbation orders, the computational complexity both for (19) and (20) is O(N 2 N x log(N x )) while the storage is O(NN x ).…”

“…So, our new computational strategy of Stabilized Field Expansions and Dirichlet-Interior Derivative Operators (SFE/DIDO) for computing Dirichlet-Neumann operators amounts to using (31) to compute the coefficients d p,n followed by the utilization of (32) to recover the normal derivative. The computational complexity of these two steps is identical to that of the original FE method (19) …”

“…For the initial value problem, Boundary Perturbations applied to the formulation of Zakharov have been very fruitful in the derivation of long-wave approximations of the Euler equations (Craig, Sulem, & Sulem [15]; Craig & Groves [18]; Craig, Guyenne, Nicholls, & Sulem [22]; and Craig, Guyenne, & Kalisch [21]), and the examination of integrability properties of the Euler equations (Craig & Worfolk [27]; Craig & Groves [19]; and Craig [17]). …”

Nonlinear Dynamics of a System of Particle-Like Wavepackets

“…Since we want to establish a general property of a wide class of systems, we apply a general enough dynamical approach. There is a number of general approaches developed for the studies of high-dimensional and infinite-dimensional nonlinear evolutionary systems of hyperbolic type, [12], [15], [21], [24], [31], [37], [42], [47], [49], [51], [53]) and references therein. The approach we develop here is based on the introduction of a wavepacket interaction system.…”