Theory of Water Waves

 Department of Mathematics
 University of Surrey

Dispersion via modulation in shallow water hydrodynamics
  A new theory for the emergence of dispersion in shallow water hydrodynamics in two horizontal space dimensions is presented. Starting with the key properties of uniform flow in open-channel hydraulics, it is shown that criticality is the key mechanism for generating dispersion. Modulation of the uniform flow then leads to model equations. The coefficients in the model equations are related precisely to the derivatives of the mass flux, momentum flux and mass density. The theory gives a new perspective -- from the viewpoint of hydraulics -- on how and why key shallow water models like the Korteweg-de Vries equation and Kadomtsev-Petviashvili equations arise in the theory of water waves.
  T.J. Bridges. Emergence of dispersion in shallow water hydrodynamics via modulation of uniform flow, J. Fluid Mech. 761 R1-R9 (2014)   JFM website    
Lectures on the theory of water waves
  In the summer of 2014 leading experts in the theory of water waves gathered at the Newton Institute for Mathematical Sciences in Cambridge for four weeks of research interaction. A cross-section of those experts was invited to give introductory-level talks on active topics. This book is a compilation of those talks and illustrates the diversity, intensity, and progress of current research in this area. The key themes that emerge are numerical methods for analysis, stability and simulation of water waves, transform methods, rigorous analysis of model equations, three-dimensionality of water waves, variational principles, shallow water hydrodynamics, the role of deterministic and random bottom topography, and modulation equations. This book is an ideal introduction for PhD students and researchers looking for a research project. It may also be used as a supplementary text for advanced courses in mathematics or fluid dynamics.
  T.J. Bridges, M.D. Groves, and D.P. Nicholls (Editors). Lectures on the Theory of Water Waves Lond. Math. Soc. Lect. Notes 426, Cambridge University Press (2016).   CUP website  

Variational principles and moving mesh
  The time-dependent motion of water waves with a parametrically-defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic PDEs. The aim is to study the effect of transformation on variational principles for water waves such as Luke's Lagrangian formulation, Zakharov's Hamiltonian formulation, and the Benjamin-Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian, and some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.
  T.J. Bridges & N.M. Donaldson. Variational principles for water waves from the viewpoint of a time-dependent moving mesh, Mathematika 57 147-173 (2011)
Mathematika webpage  

Wave breaking and the surface velocity field
  It is shown that there is an exact reduction from the full three-dimensional inviscid water wave problem with vorticity to a set of hyperbolic equations for the free surface horizontal velocity field. The key term in the reduced equations is g + A where g > 0 is the gravitational constant and A is the Lagrangian vertical acceleration at the surface. Vertical accelerations -- both Eulerian -- -- and Lagrangian -- at the surface are widely used as a diagnostic for wave breaking and this reduction gives a precise theoretical argument for how accelerations drive the surface dynamics. When the acceleration term is small, the surface equations reduce to the model equation proposed in Pomeau et al (2008 Nonlinearity 21 T61-T79, Proc. R. Soc. Lond. A 464 1851-66) which provides a universal law for spreading of the crest during wave breaking, and when the acceleration term is negative the equations can switch type (hyperbolic to elliptic) and solutions can fail to exist. Addition of surface tension leads to dispersive regularization of the surface equations.
  T.J. Bridges. Wave breaking and the surface velocity field for three-dimensional water waves, Nonlinearity 22 947-953 (2009)   Nonlinearity website    

Computation of time-dependent water wave dynamics
  Methods for the numerical computation of freely propagating irrotational water waves are reviewed. The emphasis is on the methods, not on the results. The primary focus is on methods for time-dependent fully nonlinear water waves, but aspects of steady waves are also discussed. For time-dependent waves, a range of topics from two-dimensional time-periodic waves over a flat bottom to unsteady three-dimensional waves over an arbitrary topography, including the statistical description of water waves, are discussed.
  F. Dias & T.J. Bridges. The numerical computation of freely-propagating time-dependent irrotational water waves. Fluid Dynamics Research 38 803-830 (2006)     FDR website    
Short-crested Stokes waves and two-wave interactions
  The motivation for this work is the stability problem for short-crested Stokes waves. A new point of view is proposed, based on the observation that an understanding of the linear stability of short-crested waves (SCWs) is closely associated with an understanding of the stability of the oblique non-resonant interaction between two waves. The proposed approach is to embed the SCWs in a six-parameter family of oblique non-resonant interactions. A variational framework is developed for the existence and stability of this general two-wave interaction. It is argued that the resonant SCW limit makes sense a posteriori, and leads to a new stability theory for both weakly nonlinear and finite-amplitude SCWs. Even in the weakly nonlinear case the results are new: transverse weakly nonlinear long-wave instability is independent of the nonlinear frequency correction for SCWs whereas longitudinal instability is influenced by the SCW frequency correction, and, in parameter regions of physical interest there may be more than one unstable mode. With explicit results, a critique of existing results in the literature can be given, and several errors and misconceptions in previous work are pointed out. The theory is developed in some generality for Hamiltonian PDEs. Water waves and a nonlinear wave equation in two space dimensions are used for illustration of the theory.
  T.J. Bridges & F.E. Laine-Pearson The long-wave instability of short-crested waves, via embedding in the oblique two-wave interaction, J. Fluid Mech. 543 147-182 (2005).     JFM website    

The Benjamin-Feir instability
  The existence and linear stability problem for the Stokes periodic wavetrain on fluids of finite depth is formulated in terms of the spatial and temporal Hamiltonian structure of the water-wave problem. A proof, within the Hamiltonian framework, of instability of the Stokes periodic wavetrain is presented. A Hamiltonian center-manifold analysis reduces the linear stability problem to an ordinary differential eigenvalue problem on 4D. A projection of the reduced stability problem onto the tangent space of the 2-manifold of periodic Stokes waves is used to prove the existence of a dispersion relation. A rigorous analysis of the dispersion relation proves the result, first discovered in the 1960's, that the Stokes gravity wavetrain of sufficiently small amplitude is unstable for F > F0 where F0 is approximately 0.8 and F is the Froude number. The addition of dissipation can enhance the Benjamin-Feir instability.
  T. J. Bridges & A. Meilke. A proof of the Benjamin-Feir instability, Arch. Rat. Mech. Anal. 133 145-198 (2001)     ARMA website  
  T. J. Bridges & F. Dias. Enhancement of the Benjamin-Feir instability with dissipation, Phys. Fluids 19 104104 (2007)     PoF website  

Superharmonic instability and wave breaking
  The superharmonic instability is pervasive in large-amplitude water-wave problems and numerical simulations have predicted a close connection between it and crest instabilities and wave breaking. In this paper we present a nonlinear theory, which is a generic nonlinear consequence of superharmonic instability. The theory predicts the nonlinear behaviour witnessed in numerics, and gives new information about the nonlinear structure of large-amplitude water waves, including a mechanism for noisy wave breaking.
  T.J. Bridges Superharmonic instability, homoclinic torus bifurcation and breaking water waves, J. Fluid Mech. 505 153-162 (2004)   JFM website    

Steady three-dimensional multi-periodic patterns
  The formation of doubly-periodic patterns on the surface of a fluid layer with a uniform velocity field and constant depth is considered. The fluid is assumed to be inviscid and the flow irrotational. The problem of steady patterns is shown to have a novel variational formulation, which includes a new characterization of steady uniform mean flow, and steady uniform flow coupled with steady doubly periodic patterns. A central observation is that mean flow can be characterized geometrically by associating it with symmetries. The theory gives precise information about the role of the ten natural parameters in the problem which govern the wave–mean flow interaction for steady patterns in finite depth. The formulation is applied to the problem of interaction of capillary-gravity short-crested waves with oblique travelling waves, leading to several new observations for this class of waves. Moreover, by including oblique travelling waves and short-crested waves in the same analysis, new bifurcations of short-crested waves are found, which give rise to mixed waves which may have complicated spatial structure.
  T.J. Bridges, F. Dias & D Menasce Steady three-dimensional water-wave patterns on a finite-depth fluid, J. Fluid Mech. 436 145-175 (2001).   JFM website    

Hamiltonian structure of the Kelvin-Helmholtz instability
  This two-part paper reports on various new insights into the classic Kelvin-Helmholtz 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, non-canonical 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 Kelvin-Helmholtz 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 Kelvin-Helmholtz 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 Kelvin-Helmholtz instability. Part 1: Hamiltonian structure, J. Fluid Mech. 333 301-325 (1997)   JFM website    
  T.B. Benjamin & T.J. Bridges. Reappraisal of the Kelvin-Helmholtz instability. Part 2: Interaction of the Kelvin-Helmholtz, superharmonic and Benjamin-Feir instabilities, J. Fluid Mech. 333 327-373 (1997)   JFM website    

  Department of Mathematics     University of Surrey