Mathematics of Planet Earth

Isaac Newton Institute for Mathematical Sciences

Theory of Water Waves

14 July - 8 August 2014

Organisers: Tom Bridges (Surrey), Mark Groves (Saarland), Paul Milewski (Bath) and David Nicholls (Illinois at Chicago)
TWW programme identifier: the energy density of a wave packet of capillary-gravity waves, computed using the free surface Euler equations,  at successive times. The packet focusses into a ring like feature that ultimately fragments into individual solitary structures.

Water waves are a dramatic, potentially dangerous, yet beautiful phenomena that is omnipresent and impacts every aspect of life on the planet. At smaller length scales the ripples driven by surface tension affect remote sensing. At intermediate length scales waves in the mid-ocean affect shipping and near the shoreline they control the coastal morphology and the ability to navigate along shore. At larger length scales waves such as tsunamis and hurricane-generated waves can cause devastation on a global scale. Across all length scales an exchange of momentum and thermal energy between ocean and atmosphere occurs affecting the global weather system and the climate.

From a mathematical viewpoint water waves pose rich challenges.The governing equations for water waves are a widely accepted model and they have been the subject of a wide range of research. However, the equations are highly nonlinear and the level of difficulty is so great that theory has yet to scratch the surface of the subject. The solutions to the equations that describe fluid motion are elusive and whether they even exist in the most general case is one of the most difficult unanswered questions in mathematics.

On the other hand, there is good reason to be buoyant about the headway that mathematics can make in tackling the great open problems posed by water waves. In light of recent developments the questions are now clearer, new methodologies are emerging, computational approaches are becoming much more sophisticated and the number of researchers at the highest international level involved is growing. All these indicators point to an opportune time to have a focused conference on water waves. Particular themes will include:

The initial-value problem (IVP)
  • Well-posedness, theory in two versus three dimensions,Lagrangian versus Eulerian formulations
  • Vortex sheet formulations, singularities
  • Numerical methods for the IVP, simulations
  • Theoretical aspects of wave breaking
Existence and classification of waves
  • Traveling waves in two and three dimensions
  • Periodic, multi-periodic, and localized patterns
  • Surface waves versus interfacial waves
  • Numerical methods for steady waves and their bifurcations
  • Nonlinear time-periodic waves: Standing waves and beyond
Linear and nonlinear stability of waves
  • Stability and instability of the waves mentioned above (e.g., periodic and multi-periodic traveling waves, localized traveling waves, standing waves)
  • Beyond (symmetric) short-crested waves: Stability of genuinely three-dimensional traveling waves.
  • Wave interactions
  • Rogue waves and other extreme events
  • Energetic stability
Dynamical systems and geometric techniques
  • Hamiltonian and Lagrangian formulations, variational principles
  • Center-manifold reduction, spatial dynamics, and generalizations
  • Differential geometry of the sea surface
  • The effect of bottom topography
Beyond irrotational flow
  • Waves with vorticity
  • Weakly viscous potential flow
  • Waves with forcing and dissipation