On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

The effects of surfactants on spilling breaking waves

Breaking waves markedly increase the rates of air–sea transfer of momentum, energy and mass. In light to moderate wind conditions, spilling breakers with short wavelengths are observed frequently. Theory and laboratory experiments have shown that, as these waves approach breaking in clean water, a ripple pattern that is dominated by surface tension forms at the crest. Under laboratory conditions and in theory, the transition to turbulent flow is triggered by flow separation under the ripples, typically without leading to overturning of the free surface15. Water surfaces in nature, however, are typically contaminated by surfactant films that alter the surface tension and produce surface elasticity and viscosity16, 17. Here we present the results of laboratory experiments in which spilling breaking waves were generated mechanically in water with a range of surfactant concentrations. We find significant changes in the breaking process owing to surfactants. At the highest concentration of surfactants, a small plunging jet issues from the front face of the wave at a point below the wave crest and entraps a pocket of air on impact with the front face of the wave. The bubbles and turbulence created during this process are likely to increase air–sea transfer

Geometric multigrid for an implicit-time immersed boundary method

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the structure and Eulerian variables to describe the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. These tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100--1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50--200 times more efficient than the explicit method