Surface Gravity and Capillary Waves

W.K. Melville    Scripps Institution of Oceanography, University of California, San Diego, La Jolla, USA

Introduction

Ocean surface waves are the most common oceanographic phenomena that are known to the casual observer. They can at once be the source of inspiration and primal fear. It is remarkable that the complex, random wave field of a storm-lashed sea can be studied and modeled using well-developed theoretical concepts. Many of these concepts are based on linear or weakly nonlinear approximations to the full nonlinear dynamics of ocean waves. Early contributors to these theories included such luminaries as Cauchy, Poisson, Stokes, Lagrange, Airy, Kelvin and Rayleigh. Many of the current challenges in the study of ocean surface waves are related to nonlinear processes which are not yet well understood. These include dynamical coupling between the atmosphere and the ocean, wave-wave interactions, and wave breaking.

For the purposes of this article, surface waves are considered to extend from low frequency swell from distant storms at periods of 10 s or more and wavelengths of hundreds of meters, to capillary waves with wavelengths of millimeters and frequencies of O(10) Hz. In between are wind waves with lengths of O(1-100) m and periods of O(1-10) s. Figure 1 shows a spectrum of surface waves measured from the Research Platform FLIP off the coast of Oregon. The spectrum, F, shows the distribution of energy in the wave field as a function of frequency. The wind wave peak at approximately 0.13 Hz is well separated from the swell peak at approximately 0.06 Hz.

Ocean surface waves play an important role in air-sea interaction. Momentum from the wind goes into both surface waves and currents. Ultimately the waves are dissipated either by viscosity or breaking, giving up their momentum to currents. Surface waves affect upper-ocean mixing through both wave breaking and their role in the generation of Langmuir circulations. This breaking and mixing influences the temperature of the ocean surface and thus the thermodynamics of air-sea interaction. Surface waves impose significant structural loads on ships and other structures. Remote sensing of the ocean surface, from local to global scales, depends on the surface wave field.

Basic Formulations

The dynamics and kinematics of surface waves are described by solutions of the Navier-Stokes equations for an incompressible viscous fluid, with appropriate boundary and initial conditions. Surface waves of the scale described here are usually generated by the wind, so the complete problem would include the dynamics of both the water and the air above. However, the density of the air is approximately 800 times smaller than that of the water, so many aspects of surface wave kinematics and dynamics may be considered without invoking dynamical coupling with the air above.

The influence of viscosity is represented by the Reynolds number of the flow, Re = UL/µ, where U is a characteristic velocity, L a characteristic length scale, and v = µ/P is the kinematic viscosity, where µ is the viscosity and ? the density of the fluid. The Reynolds number is the ratio of inertial forces to viscous forces in the fluid and if Re>>1, the effects of viscosity are often confined to thin boundary layers, with the interior of the fluid remaining essentially inviscid (v = 0). (This assumes a homogeneous fluid. In contrast, internal waves in a continuously stratified fluid are rotational since they introduce baroclinic generation of vorticity in the interior of the fluid). Denoting the fluid velocity by u = (u, v, w), the vorticity of the flow is given by =?×u If =0, the flow is said to be irrotational. From Kelvin's circulation theorem, the irrotational flow of an incompressible (?.u = 0) inviscid fluid will remain irrotational as the flow evolves. The essential features of surface waves may be considered in the context of incompressible irrotational flows.

For an irrotational flow, u = ?? where the scalar ? is a velocity potential. Then, by virtue of incompressibility, ? satisfies Laplace's equation

2?=0

  [1]

We denote the surface by z =(x, y,t), where (x, y) are the horizontal coordinates and t is time. The kinematic condition at the impermeable bottom at z = -h, is one of no flow through the boundary:

??z=0atz=-h

  [2]

There are two boundary conditions at z = ?:

??t+u???x+????y=w

  [3]

??t+12u2+g?=(pa-p)/?

  [4]

The first is a kinematic condition which is equivalent to imposing the condition that elements of fluid at the surface remain at the surface. The second is a dynamical condition, a Bernoulli equation, Which is equivalent to stating that the pressure p_ at z = ?_, an infinitesimal distance beneath the surface, is just a constant atmospheric pressure, pa, plus a contribution from surface tension. The effect of gravity is to impose a restoring force tending to bring the surface back to z = 0. The effect of surface tension is to reduce the curvature of the surface.

Although this formulation of surface waves is considerably simplified already, there are profound difficulties in predicting the evolution of surface waves based on these equations. Although Laplace's equation is linear, the surface boundary conditions are nonlinear and apply on a surface whose specification is a part of the solution. Our ability to accurately predict the evolution of nonlinear waves is limited and largely dependent on numerical techniques. The usual approach is to linearize the boundary conditions about z= 0.

Linear Waves

Simple harmonic surface waves are characterized by an amplitude a, half the distance between the crests and the troughs, and a wavenumber vector k with |k|= k=2p/?, where ? is the wavelength. The surface displacement, (unless otherwise stated, the real part of complex expressions is taken)

=aei(k.x-st)

  [5]

where s = 2p/T is the radian frequency and T is the wave period. Then ak is a measure of the slope of the waves, and if ak <<1, the surface boundary conditions can be linearized about z = 0.

Following linearization, the boundary conditions become

??t=w

  [6]

??t+g?=G?(?2??x2+?2??y2)atz=0

  [7]

where the linearized Laplace pressure is

a-p_=G(?2??x2+?2??y2)

  [8]

where G is the surface tension coefficient.

Substituting for ? and satisfying Laplace's equation and the boundary conditions at z= 0 and -h gives

=ig´acoshk(z+h)scoshkh

  [9]

Where

2=g´ktanhkh

  [10]

and

´=g(1+Gk2/?)

  [11]

Equations relating the frequency and wavenumber, =s(k),are known as dispersion relations, and for linear waves provide a fundamental description of the wave kinematics. The phase speed,

=s/k=(g´ktanhkh)1/2

  [12]

is the speed at which lines of constant phase (e.g., wave crests) move.

For waves propagating in the x-direction, the velocity field...