## Surface Waves

Abstract In this chapter, we first present the governing equations for linear wave theory. This provides a simple but yet powerful description of the wind-driven waves on the ocean surface. A number of important concepts are derived, including the dispersion relation. From the dispersion relation we examine some differences between waves in deep water and waves in shallow water and in particular we demonstrate that deep water waves are dispersive, while shallow water waves are non-dispersive. The wave spectrum is introduced as a convenient way to characterise the distribution of energy in the wave field and Significant Wave Height is defined. In the last section, we provide an overview of the fundamentals behind modern "third-generation" wave models. The balance equation for the wave energy spectrum is presented and the source terms are discussed. Various issues associated with operational wave forecasting systems are presented and some results of an ongoing wave forecast intercomparison project are presented. Finally, some indications of future directions for wave model research and operational systems are identified.

### 8.1 Introduction

The dominant features that we see when we look at the ocean are the wind-driven surface waves. At first sight, on a windy day, such as that shown in Fig. 8.1, the waves can look very complex, with a multitude of different scales in existence, wave breaking, sea-spray flying from the surface etc. It seems that we have a challenge before us to be able to describe the surface analytically, so that we can model and forecast the waves with any accuracy.

Centre for Australian Weather and Climate Research, Bureau of Meteorology, Melbourne, Australia e-mail: [email protected]

A. Schiller, G. B. Brassington (eds.), Operational Oceanography in the 21st Century, 203 Fig. 8.1 A typical ocean surface. (Photo courtesy Eric Schulz, Bureau of Meteorology)

However, with a number of simplifying assumptions, linear wave theory can provide a simple but yet powerful description of the wind-driven waves on the ocean surface. Although many of the assumptions may seem overly simplified at first, it will be seen that in general, linear wave theory can be used to describe many of the dominant features that we see in the wind waves. Further to this, moving forward to a statistical description of the ocean surface, we will see that modern wave models can provide very good forecasts of the sea-state, assuming that they are driven with reasonable estimates of the surface winds.

In this chapter, we first present the governing equations for the linear wave theory, and derive a few important concepts, the most important of these being the dispersion relation. From the dispersion relation we will be able to derive several interesting features of surface waves. Next, after the presentation of some basic definitions, we will provide an overview of the fundamentals behind modern "third-generation" wave models, which are currently implemented in a number of operational forecasting centres around the world.

### 8.2 Governing Equations

The treatment presented here is fairly standard and can be found with varying degrees of detail in books such as Young (1999), Holthuijsen (2007), and Kundu (1990).

Fig. 8.2 Framework for linear wave theory z = 0

In order to apply linear wave theory to the problem of surface gravity waves, we need to make a number of assumptions. These are (1) the amplitude of the waves is small compared to the wavelength and the depth of the water, (2) the depth of the water does not vary, (3) the waves are high frequency compared to the Coriolis frequency—this means we can ignore the rotation of the earth, (4) we also neglect surface tension—this means we are considering waves that are longer than about 5 cm, (5) the water is incompressible, (6) the water is of constant density, and (7) the motion is irrotational (and thus, viscosity can be ignored).

The description of the motion as "irrotational" can cause some confusion. As can be shown, the resulting velocities in this case are, in a sense, rotational, in that the water particles move in a circular pattern as the wave propagates. However, the particles do not rotate on their own axes and the resulting motion does not include shearing of the fluid, and so in a mathematical sense the motion is irrotational. Another way of saying this is that there is no vorticity.

We do not consider here the impact of the wind on the ocean surface, or the interaction of the waves with the bottom through shear stresses. Further, we will simplify the problem by only considering waves propagating in one direction (the x-direction). The free surface is described by n(x, t) and the depth of the water is H (see Fig. 8.2)

To begin with, since the motion is irrotational, we can define a velocity potential

4>(x, z, t) such that d& dd> u(x, z, t) = -1 and w(x, z, t) =— (8.1)

The governing equations are the conservation of mass and the conservation of momentum. The conservation of mass is governed by the continuity equation:

upon substitution of the velocity potential. This is known as the Laplace equation.

Given that the flow is irrotational and inviscid, the conservation of momentum is governed by the Bernoulli equation for unsteady flow:

For small amplitude waves, the velocity terms, u and w will also be small, so we may neglect the squares of these terms. Thus:

dt p

So we have two governing equations—the Laplace equation and the Bernoulli equation and we need to solve these under the constraints of specific boundary conditions. There are three relevant boundary conditions here. Firstly, we have the kinematic boundary condition of no normal flow at the bottom, i.e.

Secondly, we have the kinematic boundary condition at the surface. This dictates that fluid may not leave the surface, or in other words, the vertical velocity of the fluid is the same as the total velocity of the surface. Examination of the ocean surface depicted in Fig. 8.1 suggests that this is often not satisfied in reality. It is often possible to see sea spray departing the surface and being thrown into the air. However, we are dealing here with a simplified situation with no wind and no wave breaking. The boundary condition is:

dz Dt dt dx

Again, we note that u(dn/dx) is the product of two small terms so it may be neglected. Further, it can be shown (e.g. Kundu 1990) through a Taylor expansion of /dz that the boundary condition can be evaluated at z=0 instead of z = n. Thus the kinematic boundary condition at the free surface becomes:

dz dt

Our third boundary condition is the dynamic boundary condition at the free surface. This says that the pressure just below the surface is equal to the ambient, or atmospheric pressure. We may set this equal to anything we like, so we will set it to be equal to zero, thus:

The Bernoulli equation therefore becomes

As before, a Taylor expansion allows us to evaluate B\$/Bt at z=0 instead of at z = n so we have

Thus the problem can be stated as follows: we need to solve dlt + = 0 (8.12)

dx2 dz2

subject to d<Pdn

This is fairly straightforward to solve and a detailed solution can be found in Kundu (1990). It involves assuming a sinusoidal form for n(x, t) and using separation of variables. The solution is found to be aw cosh k(z + H) ,a

k sinh kH

where a is a constant (the wave amplitude), & is the wave frequency (m = 2nf = 2n /T where T is the wave period) and k is the wave number (k = 2n / X where X is the wavelength). From this, the velocity components u and w can readily be found using Eq. (8.1). 